hash H
hash G
name k : message
name k' : message
name s0 : index -> message
mutable sT(i:index) : message = s0(i)
mutable sR(i:index) : message = s0(i)

abstract ok : message
channel cT
channel cR

process tag(i:index) =
sT(i):=H(sT(i),k);
out(cT,G(sT(i),k'))

process reader =
in(cT,x);
try find ii such that x = G(H(sR(ii),k),k') in
sR(ii):=H(sR(ii),k);
out(cR,ok)

system (!_i !_j T: tag(i) | !_jj R: reader).
Typed-check process:

null

Added action dependencies lemmas:


System after processing:

null

System Empty registered with actions (init).
[warning>Loaded "Prelude.sp".
<]global axiom namelength_k {'P:system} @system:(set:'P; equiv:None) :
[len k = namelength_message]
global axiom namelength_k' {'P:system} @system:(set:'P; equiv:None) :
[len k' = namelength_message]
global axiom namelength_s0 {'P:system} @system:(set:'P; equiv:None) :
[forall (i:index), len (s0 i) = namelength_message]
process tag (i:index) =
sT(i) := H (sT i@τ, k); out(cT,G (sT i@τ, k')); null
process reader =
in(cT,x);
find (ii) such that x = G (H (sR ii@τ, k), k') in
sR(ii) := H (sR ii@τ, k); out(cR,ok); null
Typed-check process:

( !_i( !_j( T: tag i)) ) | !_jj( R: reader )

Added action dependencies lemmas:

axiom mutex_R1_R {'P:system[like default]} @system:(set:'P; equiv:None) :
forall (jj,ii:index), not happens(R1(jj)) || not happens(R(jj, ii))
axiom mutex_R_R1 {'P:system[like default]} @system:(set:'P; equiv:None) :
forall (jj,ii:index), not happens(R(jj, ii)) || not happens(R1(jj))
axiom depends_init_R1 {'P:system[like default]} @system:(set:'P; equiv:None)
: forall (jj:index), happens(R1(jj)) => init < R1(jj)
axiom depends_init_R {'P:system[like default]} @system:(set:'P; equiv:None) :
forall (jj,ii:index), happens(R(jj, ii)) => init < R(jj, ii)
axiom depends_init_T {'P:system[like default]} @system:(set:'P; equiv:None) :
forall (i,j:index), happens(T(i, j)) => init < T(i, j)

System after processing:

( !_i( !_j( sT(i) := H (sT i@τ, k); T: out(cT,G (sT i@τ, k')); null)) ) |
!_jj(
in(cT,x);
find (ii) such that x = G (H (sR ii@τ, k), k') in
sR(ii) := H (sR ii@τ, k); R: out(cR,ok); null
else
R1: null)

System Empty registered with actions (init).
System default registered with actions (init,T,R,R1).

RUNNING EXAMPLE

This protocol is a variant of the OSK protocol described in: M. Ohkubo, K. Suzuki, S. Kinoshita et al., “Cryptographic approach to “privacy-friendly” tags,” RFID privacy workshop, vol. 82. Cambridge, USA, 2003.

Each tag is associated to a mutable state sT initialized with s0. Readers have access to a database containing an entry sR for each authorized tag.

sT := H(sT,k) T -> R : G(sT,k’)

input x; sR := H(sR,k) if x = G(H(sR,k),k’) with sR in DB R -> T : ok

COMMENTS - In this model we use two keyed hash functions H and G with fixed keys k and k’, instead of two not keyed hash functions as in the specification. We address this issue in the file running-ex-oracle.sp by adding a process in order to provide the attacker the ability to compute hashes with their respective keys (without knowing these keys).

HELPING LEMMAS - last update - disjoint chains

SECURITY PROPERTIES - authentication ******************************************************************************



include Core.
op assoc ['a] (f:'a -> 'a -> 'a) : bool =
forall (x,y,z:'a), f (f x y) z = f x (f y z)
axiom eq_iff {'P:system} @system:(set:'P; equiv:None) :
forall (x,y:bool), (x = y) = (x <=> y)
axiom eq_not {'P:system} @system:(set:'P; equiv:None) :
forall (x,y:bool), (not x = not y) = (x = y)
Goal eq_sym :
(x = y) = (y = x)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
----------------------------------------
(x = y) = (y = x)

[> Line 12: by (rewrite) [goal> lemma eq_sym is proved

lemma eq_sym {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), (x = y) = (y = x)
Exiting proof mode.

Goal neq_sym :
(x <> y) = (y <> x)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
----------------------------------------
(x <> y) = (y <> x)

[> Line 15: by (rewrite) [goal> lemma neq_sym is proved

lemma neq_sym {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), (x <> y) = (y <> x)
Exiting proof mode.

Goal eq_refl_e :
(x = x) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x:'a
----------------------------------------
(x = x) = true

[> Line 20: by (rewrite) [goal> lemma eq_refl_e is proved

lemma eq_refl_e {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x:'a), (x = x) = true
Exiting proof mode.

Goal eq_refl :
x = x
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x:'a
----------------------------------------
x = x

[> Line 26: by (rewrite) [goal> lemma eq_refl is proved

lemma eq_refl {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x:'a), x = x
Exiting proof mode.

Goal neq_irrefl :
x <> x <=> false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x:'a
----------------------------------------
x <> x <=> false

[> Line 29: by (split) [goal> lemma neq_irrefl is proved

lemma neq_irrefl {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x:'a), x <> x <=> false
Exiting proof mode.

Goal eq_assoc :
((b0 = b1) = b2) = (b0 = (b1 = b2))
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1,b2:bool
----------------------------------------
((b0 = b1) = b2) = (b0 = (b1 = b2))

[> Line 37: ((have); 1: by (rewrite)) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1,b2:bool[const]
true_false: (true = false) = false
----------------------------------------
((b0 = b1) = b2) = (b0 = (b1 = b2))

[> Line 38: ((have); 1: by (rewrite)) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1,b2:bool[const]
false_true: (false = true) = false
true_false: (true = false) = false
----------------------------------------
((b0 = b1) = b2) = (b0 = (b1 = b2))

[> Line 38: ((case);((case);((case);(try (auto))))) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1,b2:bool[const]
false_true: (false = true) = false
true_false: (true = false) = false
----------------------------------------
not b2 => not b1 => b0 => ((true = false) = false) = (true = (false = false))

[> Line 38: by (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1,b2:bool[const]
false_true: (false = true) = false
true_false: (true = false) = false
----------------------------------------
b2 => not b1 => not b0 => ((false = false) = true) = (false = (false = true))

[> Line 38: by (rewrite) [goal> lemma eq_assoc is proved

lemma eq_assoc {'P:system} @system:(set:'P; equiv:None) :
forall (b0,b1,b2:bool), ((b0 = b1) = b2) = (b0 = (b1 = b2))
Exiting proof mode.

axiom fun_ext {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
forall (f,g:'a -> 'b), (forall (x:'a), f x = g x) => f = g
Goal true_false :
(true = false) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
----------------------------------------
(true = false) = false

[> Line 51: by (rewrite) [goal> lemma true_false is proved

lemma true_false {'P:system} @system:(set:'P; equiv:None) :
(true = false) = false
Exiting proof mode.

Goal false_true :
(false = true) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
----------------------------------------
(false = true) = false

[> Line 57: by (rewrite) [goal> lemma false_true is proved

lemma false_true {'P:system} @system:(set:'P; equiv:None) :
(false = true) = false
Exiting proof mode.

Goal eq_true :
(b = true) = b
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(b = true) = b

[> Line 61: by (case) [goal> lemma eq_true is proved

lemma eq_true {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (b = true) = b
Exiting proof mode.

Goal eq_true2 :
(true = b) = b
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(true = b) = b

[> Line 65: by (case) [goal> lemma eq_true2 is proved

lemma eq_true2 {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (true = b) = b
Exiting proof mode.

axiom not_true {'P:system} @system:(set:'P; equiv:None) : not true = false
axiom not_false {'P:system} @system:(set:'P; equiv:None) : not false = true
Goal not_not :
not (not b) = b
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
not (not b) = b

[> Line 81: by (case) [goal> lemma not_not is proved

lemma not_not {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), not (not b) = b
Exiting proof mode.

Goal not_eq :
not (x = y) = (x <> y)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
----------------------------------------
not (x = y) = (x <> y)

[> Line 86: by (rewrite) [goal> lemma not_eq is proved

lemma not_eq {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), not (x = y) = (x <> y)
Exiting proof mode.

Goal not_neq :
not (x <> y) = (x = y)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
----------------------------------------
not (x <> y) = (x = y)

[> Line 92: by (rewrite) [goal> lemma not_neq is proved

lemma not_neq {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), not (x <> y) = (x = y)
Exiting proof mode.

Goal not_eqfalse :
(b = false) = not b
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(b = false) = not b

[> Line 99: by (case) [goal> lemma not_eqfalse is proved

lemma not_eqfalse {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (b = false) = not b
Exiting proof mode.

Goal not_impl :
not (a => b) = (a && not b)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool
----------------------------------------
not (a => b) = (a && not b)

[> Line 104: ((rewrite);((split);(intro))) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
H: not (a => b)
----------------------------------------
a && not b

[> Line 104: (split) [goal> Focused goal (1/3):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
H: not (a => b)
----------------------------------------
a

[> Line 106: (rewrite) [goal> Focused goal (1/3):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
H: not (a => b)
----------------------------------------
not (not a)

[> Line 107: (intro) [goal> Focused goal (1/3):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
H: not (a => b)
Hna: not a
----------------------------------------
false

[> Line 108: by (apply) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
H: not (a => b)
----------------------------------------
not b

[> Line 109: (intro) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
H: not (a => b)
Hb: b
----------------------------------------
false

[> Line 110: by (apply) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
H: a && not b
----------------------------------------
not (a => b)

[> Line 111: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
H: a && not b
Hi: a => b
----------------------------------------
false

[> Line 112: (destruct) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
Ha: a
Hi: a => b
Hnb: not b
----------------------------------------
false

[> Line 113: (apply) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
Ha: a
Hi: a => b
Hnb: not b
----------------------------------------
b

[> Line 114: by (apply) [goal> lemma not_impl is proved

lemma not_impl {'P:system} @system:(set:'P; equiv:None) :
forall (a,b:bool), not (a => b) = (a && not b)
Exiting proof mode.

Goal eq_false :
((x = y) = false) = (x <> y)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
----------------------------------------
((x = y) = false) = (x <> y)

[> Line 121: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
----------------------------------------
((x = y) = false) = not (x = y)

[> Line 121: ((case);(intro)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
_: x = y
----------------------------------------
(true = false) = not true

[> Line 121: (simpl) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
_: x = y
----------------------------------------
true

[> Line 122: (auto) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
_: not (x = y)
----------------------------------------
(false = false) = not false

[> Line 122: by (rewrite) [goal> lemma eq_false is proved

lemma eq_false {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), ((x = y) = false) = (x <> y)
Exiting proof mode.

axiom and_comm {'P:system} @system:(set:'P; equiv:None) :
forall (b,b':bool), (b && b') = (b' && b)
Goal and_dist :
((b0 || b1) && b2) = (b0 && b2 || b1 && b2)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1,b2:bool
----------------------------------------
((b0 || b1) && b2) = (b0 && b2 || b1 && b2)

[> Line 132: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1,b2:bool[const]
----------------------------------------
(b0 || b1) && b2 <=> b0 && b2 || b1 && b2

[> Line 132: by (split) [goal> lemma and_dist is proved

lemma and_dist {'P:system} @system:(set:'P; equiv:None) :
forall (b0,b1,b2:bool), ((b0 || b1) && b2) = (b0 && b2 || b1 && b2)
Exiting proof mode.

axiom and_true_l {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (true && b) = b
Goal and_true_r :
(b && true) = b
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(b && true) = b

[> Line 138: by (rewrite) [goal> lemma and_true_r is proved

lemma and_true_r {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (b && true) = b
Exiting proof mode.

axiom and_false_l {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (false && b) = false
Goal and_false_r :
(b && false) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(b && false) = false

[> Line 145: by (rewrite) [goal> lemma and_false_r is proved

lemma and_false_r {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (b && false) = false
Exiting proof mode.

Goal and_double :
(b && b) = b
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(b && b) = b

[> Line 150: by (case) [goal> lemma and_double is proved

lemma and_double {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (b && b) = b
Exiting proof mode.

axiom or_comm {'P:system} @system:(set:'P; equiv:None) :
forall (b,b':bool), (b || b') = (b' || b)
Goal or_dist :
((b0 || b2) && (b1 || b2)) = (b0 && b1 || b2)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1,b2:bool
----------------------------------------
((b0 || b2) && (b1 || b2)) = (b0 && b1 || b2)

[> Line 158: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1,b2:bool[const]
----------------------------------------
(b0 || b2) && (b1 || b2) <=> b0 && b1 || b2

[> Line 158: by (split) [goal> lemma or_dist is proved

lemma or_dist {'P:system} @system:(set:'P; equiv:None) :
forall (b0,b1,b2:bool), ((b0 || b2) && (b1 || b2)) = (b0 && b1 || b2)
Exiting proof mode.

axiom or_false_l {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (false || b) = b
Goal or_false_r :
(b || false) = b
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(b || false) = b

[> Line 164: by (rewrite) [goal> lemma or_false_r is proved

lemma or_false_r {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (b || false) = b
Exiting proof mode.

axiom or_true_l {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (true || b) = true
Goal or_true_r :
(b || true) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(b || true) = true

[> Line 171: by (rewrite) [goal> lemma or_true_r is proved

lemma or_true_r {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (b || true) = true
Exiting proof mode.

Goal or_double :
(b || b) = b
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(b || b) = b

[> Line 175: by (case) [goal> lemma or_double is proved

lemma or_double {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (b || b) = b
Exiting proof mode.

Goal impl_charac :
(b => b') = (not b || b')
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b,b':bool
----------------------------------------
(b => b') = (not b || b')

[> Line 182: (((rewrite);((split);((case);(case))));(intro)) [goal> lemma impl_charac is proved

lemma impl_charac {'P:system} @system:(set:'P; equiv:None) :
forall (b,b':bool), (b => b') = (not b || b')
Exiting proof mode.

Goal impl_false_l :
(false => b) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(false => b) = true

[> Line 186: by ((rewrite);(case)) [goal> lemma impl_false_l is proved

lemma impl_false_l {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (false => b) = true
Exiting proof mode.

Goal impl_true_r :
(b => true) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(b => true) = true

[> Line 190: (auto) [goal> lemma impl_true_r is proved

lemma impl_true_r {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (b => true) = true
Exiting proof mode.

Goal impl_true_l :
(true => b) = b
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool
----------------------------------------
(true => b) = b

[> Line 194: by (rewrite) [goal> lemma impl_true_l is proved

lemma impl_true_l {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool), (true => b) = b
Exiting proof mode.

Goal impl_contra :
(b => c) = (not c => not b)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b,c:bool
----------------------------------------
(b => c) = (not c => not b)

[> Line 200: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b,c:bool[const]
----------------------------------------
(not b || c) = (c || not b)

[> Line 201: by (rewrite) [goal> lemma impl_contra is proved

lemma impl_contra {'P:system} @system:(set:'P; equiv:None) :
forall (b,c:bool), (b => c) = (not c => not b)
Exiting proof mode.

Goal not_and :
not (a && b) = (not a || not b)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool
----------------------------------------
not (a && b) = (not a || not b)

[> Line 208: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
----------------------------------------
not (a && b) <=> not a || not b

[> Line 210: (((case);(case));(intro)) [goal> lemma not_and is proved

lemma not_and {'P:system} @system:(set:'P; equiv:None) :
forall (a,b:bool), not (a && b) = (not a || not b)
Exiting proof mode.

Goal not_or :
not (a || b) = (not a && not b)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool
----------------------------------------
not (a || b) = (not a && not b)

[> Line 214: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: a,b:bool[const]
----------------------------------------
not (a || b) <=> not a && not b

[> Line 216: (((case);(case));(intro)) [goal> lemma not_or is proved

lemma not_or {'P:system} @system:(set:'P; equiv:None) :
forall (a,b:bool), not (a || b) = (not a && not b)
Exiting proof mode.

Goal if_true :
b => if b then x else y = x
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool,x,y:'a
----------------------------------------
b => if b then x else y = x

[> Line 225: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool[const],x,y:'a
H: b
----------------------------------------
if b then x else y = x

[> Line 226: (case) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool[const],x,y:'a
H: b
----------------------------------------
b && if b then x else y = x => x = x

[> Line 227: (auto) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool[const],x,y:'a
H: b
----------------------------------------
not b && if b then x else y = y => y = x

[> Line 227: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool[const],x,y:'a
H: b
HH: not b
_: if b then x else y = y
----------------------------------------
y = x

[> Line 228: by (have) [goal> lemma if_true is proved

lemma if_true {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (b:bool,x,y:'a), b => if b then x else y = x
Exiting proof mode.

Goal if_true0 :
if true then x else y = x
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
----------------------------------------
if true then x else y = x

[> Line 233: by (rewrite) [goal> lemma if_true0 is proved

lemma if_true0 {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), if true then x else y = x
Exiting proof mode.

Goal if_false :
not b => if b then x else y = y
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool,x,y:'a
----------------------------------------
not b => if b then x else y = y

[> Line 241: ((intro);(case)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool[const],x,y:'a
H: not b
----------------------------------------
b && if b then x else y = x => x = y

[> Line 241: (intro) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool[const],x,y:'a
H: not b
H1: b
H2: if b then x else y = x
----------------------------------------
x = y

[> Line 242: by (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool[const],x,y:'a
H: not b
----------------------------------------
not b && if b then x else y = y => y = y

[> Line 244: (auto) [goal> lemma if_false is proved

lemma if_false {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (b:bool,x,y:'a), not b => if b then x else y = y
Exiting proof mode.

Goal if_false0 :
if false then x else y = y
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
----------------------------------------
if false then x else y = y

[> Line 250: by (rewrite) [goal> lemma if_false0 is proved

lemma if_false0 {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), if false then x else y = y
Exiting proof mode.

Goal if_then_then :
if b then (if b' then x else y) else y = if (b && b') then x else y
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b,b':bool,x,y:'a
----------------------------------------
if b then (if b' then x else y) else y = if (b && b') then x else y

[> Line 257: by ((case);(case)) [goal> lemma if_then_then is proved

lemma if_then_then {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (b,b':bool,x,y:'a),
if b then (if b' then x else y) else y = if (b && b') then x else y
Exiting proof mode.

Goal if_then_or :
if b0 then m0 else if b1 then m0 else m1 = if (b0 || b1) then m0 else m1
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1:bool,m0,m1:message
----------------------------------------
if b0 then m0 else if b1 then m0 else m1 = if (b0 || b1) then m0 else m1

[> Line 264: ((have); 1: by (auto)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1:bool[const],m0,m1:message
_: b0
----------------------------------------
if b0 then m0 else if b1 then m0 else m1 = if (b0 || b1) then m0 else m1

[> Line 264: ((rewrite);(intro)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1:bool[const],m0,m1:message
_: b0
----------------------------------------
m0 = if (b0 || b1) then m0 else m1

[> Line 264: ((rewrite);(intro)) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1:bool[const],m0,m1:message
_: not b0
----------------------------------------
if b0 then m0 else if b1 then m0 else m1 = if (b0 || b1) then m0 else m1

[> Line 265: ((have); 1: by (auto)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1:bool[const],m0,m1:message
_: b1
_: not b0
----------------------------------------
if b0 then m0 else if b1 then m0 else m1 = if (b0 || b1) then m0 else m1

[> Line 265: ((rewrite);(intro)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1:bool[const],m0,m1:message
_: b1
_: not b0
----------------------------------------
if b1 then m0 else m1 = if (b0 || b1) then m0 else m1

[> Line 265: ((rewrite);(intro)) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b0,b1:bool[const],m0,m1:message
_: not b1
_: not b0
----------------------------------------
if b0 then m0 else if b1 then m0 else m1 = if (b0 || b1) then m0 else m1

[> Line 267: ((rewrite);(intro)) [goal> lemma if_then_or is proved

lemma if_then_or {'P:system} @system:(set:'P; equiv:None) :
forall (b0,b1:bool,m0,m1:message),
if b0 then m0 else if b1 then m0 else m1 = if (b0 || b1) then m0 else m1
Exiting proof mode.

Goal if_then_implies :
if b then (if b' then x else y) else z =
if b then (if (b => b') then x else y) else z
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b,b':bool,x,y,z:'a
----------------------------------------
if b then (if b' then x else y) else z =
if b then (if (b => b') then x else y) else z

[> Line 273: ((case);((intro);((case);((intro);((simpl);(try (auto))))))) [goal> lemma if_then_implies is proved

lemma if_then_implies {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (b,b':bool,x,y,z:'a),
if b then (if b' then x else y) else z =
if b then (if (b => b') then x else y) else z
Exiting proof mode.

Goal if_same :
if b then x else x = x
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b:bool,x:'a
----------------------------------------
if b then x else x = x

[> Line 280: by (case) [goal> lemma if_same is proved

lemma if_same {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (b:bool,x:'a), if b then x else x = x
Exiting proof mode.

Goal if_then :
b = b' => if b then (if b' then x else y) else z = if b then x else z
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b,b':bool,x,y,z:'a
----------------------------------------
b = b' => if b then (if b' then x else y) else z = if b then x else z

[> Line 289: by ((intro);(case)) [goal> lemma if_then is proved

lemma if_then {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (b,b':bool,x,y,z:'a),
b = b' => if b then (if b' then x else y) else z = if b then x else z
Exiting proof mode.

Goal if_then_inv :
if b then m0 else m1 = if b then (if b then m0) else m1
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool,m0,m1:message
----------------------------------------
if b then m0 else m1 = if b then (if b then m0) else m1

[> Line 295: (auto) [goal> lemma if_then_inv is proved

lemma if_then_inv {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool,m0,m1:message),
if b then m0 else m1 = if b then (if b then m0) else m1
Exiting proof mode.

Goal if_else :
b = b' => if b then x else if b' then y else z = if b then x else z
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b,b':bool,x,y,z:'a
----------------------------------------
b = b' => if b then x else if b' then y else z = if b then x else z

[> Line 303: by ((intro);(case)) [goal> lemma if_else is proved

lemma if_else {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (b,b':bool,x,y,z:'a),
b = b' => if b then x else if b' then y else z = if b then x else z
Exiting proof mode.

Goal if_else_inv :
if b then m0 else m1 = if b then m0 else if not b then m1
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool,m0,m1:message
----------------------------------------
if b then m0 else m1 = if b then m0 else if not b then m1

[> Line 308: by (case) [goal> lemma if_else_inv is proved

lemma if_else_inv {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool,m0,m1:message),
if b then m0 else m1 = if b then m0 else if not b then m1
Exiting proof mode.

Goal if_push :
if b then m0 else m1 = if b then (if b then m0) else if not b then m1
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: b:bool,m0,m1:message
----------------------------------------
if b then m0 else m1 = if b then (if b then m0) else if not b then m1

[> Line 312: by (rewrite) [goal> lemma if_push is proved

lemma if_push {'P:system} @system:(set:'P; equiv:None) :
forall (b:bool,m0,m1:message),
if b then m0 else m1 = if b then (if b then m0) else if not b then m1
Exiting proof mode.

Goal if_then_not :
b = not b' => if b then (if b' then x else y) else z = if b then y else z
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b,b':bool,x,y,z:'a
----------------------------------------
b = not b' => if b then (if b' then x else y) else z = if b then y else z

[> Line 320: by ((intro);(case)) [goal> lemma if_then_not is proved

lemma if_then_not {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (b,b':bool,x,y,z:'a),
b = not b' => if b then (if b' then x else y) else z = if b then y else z
Exiting proof mode.

Goal if_else_not :
b = not b' => if b then x else if b' then y else z = if b then x else y
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: b,b':bool,x,y,z:'a
----------------------------------------
b = not b' => if b then x else if b' then y else z = if b then x else y

[> Line 329: by ((intro);(case)) [goal> lemma if_else_not is proved

lemma if_else_not {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (b,b':bool,x,y,z:'a),
b = not b' => if b then x else if b' then y else z = if b then x else y
Exiting proof mode.

Goal if_app :
f (if c then x else y) = if c then f x else f y
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: c:bool,f:'a -> 'b,x,y:'a
----------------------------------------
f (if c then x else y) = if c then f x else f y

[> Line 334: by (case) [goal> lemma if_app is proved

lemma if_app {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
forall (f:'a -> 'b,c:bool,x,y:'a),
f (if c then x else y) = if c then f x else f y
Exiting proof mode.

Goal fst_pair :
fst <x,y> = x
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x,y:message
----------------------------------------
fst <x,y> = x

[> Line 340: (auto) [goal> lemma fst_pair is proved

lemma fst_pair {'P:system} @system:(set:'P; equiv:None) :
forall (x,y:message), fst <x,y> = x
Exiting proof mode.

Goal snd_pair :
snd <x,y> = y
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x,y:message
----------------------------------------
snd <x,y> = y

[> Line 344: (auto) [goal> lemma snd_pair is proved

lemma snd_pair {'P:system} @system:(set:'P; equiv:None) :
forall (x,y:message), snd <x,y> = y
Exiting proof mode.

Goal iff_def :
(x <=> y) = ((x => y) && (y => x))
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x,y:bool
----------------------------------------
(x <=> y) = ((x => y) && (y => x))

[> Line 353: ((rewrite);(split)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x,y:bool[const]
----------------------------------------
x <=> y => (x => y) && (y => x)

[> Line 353: by (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x,y:bool[const]
----------------------------------------
(x => y) && (y => x) => x <=> y

[> Line 355: (auto) [goal> lemma iff_def is proved

lemma iff_def {'P:system} @system:(set:'P; equiv:None) :
forall (x,y:bool), (x <=> y) = ((x => y) && (y => x))
Exiting proof mode.

Goal iff_refl :
(x <=> x) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x:bool
----------------------------------------
(x <=> x) = true

[> Line 359: by (rewrite) [goal> lemma iff_refl is proved

lemma iff_refl {'P:system} @system:(set:'P; equiv:None) :
forall (x:bool), (x <=> x) = true
Exiting proof mode.

Goal iff_sym :
(x <=> y) = (y <=> x)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x,y:bool
----------------------------------------
(x <=> y) = (y <=> x)

[> Line 366: by (rewrite) [goal> lemma iff_sym is proved

lemma iff_sym {'P:system} @system:(set:'P; equiv:None) :
forall (x,y:bool), (x <=> y) = (y <=> x)
Exiting proof mode.

Goal true_iff_false :
(true <=> false) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
----------------------------------------
(true <=> false) = false

[> Line 370: by (rewrite) [goal> lemma true_iff_false is proved

lemma true_iff_false {'P:system} @system:(set:'P; equiv:None) :
(true <=> false) = false
Exiting proof mode.

Goal false_iff_true :
(false <=> true) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
----------------------------------------
(false <=> true) = false

[> Line 376: by (rewrite) [goal> lemma false_iff_true is proved

lemma false_iff_true {'P:system} @system:(set:'P; equiv:None) :
(false <=> true) = false
Exiting proof mode.

Goal contra_iff :
(not x <=> y) = (x <=> not y)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x,y:bool
----------------------------------------
(not x <=> y) = (x <=> not y)

[> Line 384: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x,y:bool[const]
----------------------------------------
(not x <=> y) <=> (x <=> not y)

[> Line 385: ((split);by (rewrite)) [goal> lemma contra_iff is proved

lemma contra_iff {'P:system} @system:(set:'P; equiv:None) :
forall (x,y:bool), (not x <=> y) = (x <=> not y)
Exiting proof mode.

Goal exists_false1 :
(exists (a:'a), false) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
----------------------------------------
(exists (a:'a), false) = false

[> Line 392: by (rewrite) [goal> lemma exists_false1 is proved

lemma exists_false1 {'P:system} @system:(set:'P; equiv:None) ['a] :
(exists (a:'a), false) = false
Exiting proof mode.

Goal exists_false2 :
(exists (a:'a,b:'b), false) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
----------------------------------------
(exists (a:'a,b:'b), false) = false

[> Line 396: by (rewrite) [goal> lemma exists_false2 is proved

lemma exists_false2 {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
(exists (a:'a,b:'b), false) = false
Exiting proof mode.

Goal exists_false3 :
(exists (a:'a,b:'b,c:'c), false) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b, 'c
----------------------------------------
(exists (a:'a,b:'b,c:'c), false) = false

[> Line 400: by (rewrite) [goal> lemma exists_false3 is proved

lemma exists_false3 {'P:system} @system:(set:'P; equiv:None) ['a 'b 'c] :
(exists (a:'a,b:'b,c:'c), false) = false
Exiting proof mode.

Goal exists_false4 :
(exists (a:'a,b:'b,c:'c,d:'d), false) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b, 'c, 'd
----------------------------------------
(exists (a:'a,b:'b,c:'c,d:'d), false) = false

[> Line 404: by (rewrite) [goal> lemma exists_false4 is proved

lemma exists_false4 {'P:system} @system:(set:'P; equiv:None) ['a 'b 'c 'd] :
(exists (a:'a,b:'b,c:'c,d:'d), false) = false
Exiting proof mode.

Goal exists_false5 :
(exists (a:'a,b:'b,c:'c,d:'d,e:'e), false) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b, 'c, 'd, 'e
----------------------------------------
(exists (a:'a,b:'b,c:'c,d:'d,e:'e), false) = false

[> Line 408: by (rewrite) [goal> lemma exists_false5 is proved

lemma exists_false5 {'P:system} @system:(set:'P; equiv:None)
['a 'b 'c 'd 'e] : (exists (a:'a,b:'b,c:'c,d:'d,e:'e), false) = false
Exiting proof mode.

Goal exists_false6 :
(exists (a:'a,b:'b,c:'c,d:'d,e:'e,f:'f), false) = false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b, 'c, 'd, 'e, 'f
----------------------------------------
(exists (a:'a,b:'b,c:'c,d:'d,e:'e,f:'f), false) = false

[> Line 412: by (rewrite) [goal> lemma exists_false6 is proved

lemma exists_false6 {'P:system} @system:(set:'P; equiv:None)
['a 'b 'c 'd 'e 'f] :
(exists (a:'a,b:'b,c:'c,d:'d,e:'e,f:'f), false) = false
Exiting proof mode.

Goal forall_true1 :
(forall (a:'a), true) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
----------------------------------------
(forall (a:'a), true) = true

[> Line 422: (auto) [goal> lemma forall_true1 is proved

lemma forall_true1 {'P:system} @system:(set:'P; equiv:None) ['a] :
(forall (a:'a), true) = true
Exiting proof mode.

Goal forall_true2 :
(forall (a:'a,b:'b), true) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
----------------------------------------
(forall (a:'a,b:'b), true) = true

[> Line 426: (auto) [goal> lemma forall_true2 is proved

lemma forall_true2 {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
(forall (a:'a,b:'b), true) = true
Exiting proof mode.

Goal forall_true3 :
(forall (a:'a,b:'b,c:'c), true) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b, 'c
----------------------------------------
(forall (a:'a,b:'b,c:'c), true) = true

[> Line 430: (auto) [goal> lemma forall_true3 is proved

lemma forall_true3 {'P:system} @system:(set:'P; equiv:None) ['a 'b 'c] :
(forall (a:'a,b:'b,c:'c), true) = true
Exiting proof mode.

Goal forall_true4 :
(forall (a:'a,b:'b,c:'c,d:'d), true) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b, 'c, 'd
----------------------------------------
(forall (a:'a,b:'b,c:'c,d:'d), true) = true

[> Line 434: (auto) [goal> lemma forall_true4 is proved

lemma forall_true4 {'P:system} @system:(set:'P; equiv:None) ['a 'b 'c 'd] :
(forall (a:'a,b:'b,c:'c,d:'d), true) = true
Exiting proof mode.

Goal forall_true5 :
(forall (a:'a,b:'b,c:'c,d:'d,e:'e), true) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b, 'c, 'd, 'e
----------------------------------------
(forall (a:'a,b:'b,c:'c,d:'d,e:'e), true) = true

[> Line 438: (auto) [goal> lemma forall_true5 is proved

lemma forall_true5 {'P:system} @system:(set:'P; equiv:None)
['a 'b 'c 'd 'e] : (forall (a:'a,b:'b,c:'c,d:'d,e:'e), true) = true
Exiting proof mode.

Goal forall_true6 :
(forall (a:'a,b:'b,c:'c,d:'d,e:'e,f:'f), true) = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b, 'c, 'd, 'e, 'f
----------------------------------------
(forall (a:'a,b:'b,c:'c,d:'d,e:'e,f:'f), true) = true

[> Line 442: (auto) [goal> lemma forall_true6 is proved

lemma forall_true6 {'P:system} @system:(set:'P; equiv:None)
['a 'b 'c 'd 'e 'f] : (forall (a:'a,b:'b,c:'c,d:'d,e:'e,f:'f), true) = true
Exiting proof mode.

axiom len_zeroes {'P:system} @system:(set:'P; equiv:None) :
forall (x:message), len (zeroes x) = len x
Goal f_apply :
x = y => f x = f y
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: f:'a -> 'b,x,y:'a
----------------------------------------
x = y => f x = f y

[> Line 455: by (intro) [goal> lemma f_apply is proved

lemma f_apply {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
forall (f:'a -> 'b,x,y:'a), x = y => f x = f y
Exiting proof mode.

Goal not_exists_1 :
not exists (a:'a), phi a = forall (a:'a), not (phi a)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool
----------------------------------------
not exists (a:'a), phi a = forall (a:'a), not (phi a)

[> Line 461: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool
----------------------------------------
not exists (a:'a), phi a <=> forall (a:'a), not (phi a)

[> Line 462: (split) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool
----------------------------------------
not exists (a:'a), phi a => forall (a:'a), not (phi a)

[> Line 463: (intro) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: a:'a,phi:'a -> bool
H: not exists (a:'a), phi a
Hp: phi a
----------------------------------------
false

[> Line 463: (apply) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: a:'a,phi:'a -> bool
H: not exists (a:'a), phi a
Hp: phi a
----------------------------------------
exists (a:'a), phi a

[> Line 465: by (exists) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool
----------------------------------------
(forall (a:'a), not (phi a)) => not exists (a:'a), phi a

[> Line 465: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: a:'a,phi:'a -> bool
H: forall (a:'a), not (phi a)
Hp: phi a
----------------------------------------
false

[> Line 467: by (have) [goal> lemma not_exists_1 is proved

lemma not_exists_1 {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (phi:'a -> bool),
not exists (a:'a), phi a = forall (a:'a), not (phi a)
Exiting proof mode.

Goal not_exists_2 :
not exists (a:'a,b:'b), phi a b = forall (a:'a,b:'b), not (phi a b)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool
----------------------------------------
not exists (a:'a,b:'b), phi a b = forall (a:'a,b:'b), not (phi a b)

[> Line 473: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool
----------------------------------------
not exists (a:'a,b:'b), phi a b <=> forall (a:'a,b:'b), not (phi a b)

[> Line 474: (split) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool
----------------------------------------
not exists (a:'a,b:'b), phi a b => forall (a:'a,b:'b), not (phi a b)

[> Line 475: (intro) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: a:'a,b:'b,phi:'a -> 'b -> bool
H: not exists (a:'a,b:'b), phi a b
Hp: phi a b
----------------------------------------
false

[> Line 475: (apply) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: a:'a,b:'b,phi:'a -> 'b -> bool
H: not exists (a:'a,b:'b), phi a b
Hp: phi a b
----------------------------------------
exists (a:'a,b:'b), phi a b

[> Line 477: by (exists) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool
----------------------------------------
(forall (a:'a,b:'b), not (phi a b)) => not exists (a:'a,b:'b), phi a b

[> Line 477: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: a:'a,b:'b,phi:'a -> 'b -> bool
H: forall (a:'a,b:'b), not (phi a b)
Hp: phi a b
----------------------------------------
false

[> Line 479: by (have) [goal> lemma not_exists_2 is proved

lemma not_exists_2 {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
forall (phi:'a -> 'b -> bool),
not exists (a:'a,b:'b), phi a b = forall (a:'a,b:'b), not (phi a b)
Exiting proof mode.

axiom not_forall_1 {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (phi:'a -> bool),
not forall (a:'a), phi a = exists (a:'a), not (phi a)
axiom not_forall_2 {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
forall (phi:'a -> 'b -> bool),
not forall (a:'a,b:'b), phi a b = exists (a:'a,b:'b), not (phi a b)
axiom try_carac_1 {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
forall (phi:'a -> bool,f:'a -> 'b,g:'b),
try find x:'a such that phi x in f x else g =
if (exists (x:'a), phi x) then f (choose phi) else g
Goal choose_spec :
phi x => phi (choose phi)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
----------------------------------------
phi x => phi (choose phi)

[> Line 505: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
H: phi x
----------------------------------------
phi (choose phi)

[> Line 508: (have) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
H: phi x
----------------------------------------
phi (choose phi) = if (exists (x:'a), phi x) then phi (choose phi) else false

[> Line 509: ?? [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
H: phi x
----------------------------------------
phi (choose phi) = if (exists (x:'a), phi x) then phi (choose phi) else false

[> Line 509: ((rewrite);(intro)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
H: phi x
----------------------------------------
exists (x:'a), phi x

[> Line 510: by (exists) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
H: phi x
----------------------------------------
if (exists (x:'a), phi x) then phi (choose phi) else false

[> Line 511: ?? [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
H: phi x
----------------------------------------
if (exists (x:'a), phi x) then phi (choose phi) else false

[> Line 512: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
H: phi x
----------------------------------------
try find x:'a such that phi x in phi x else false

[> Line 513: (case) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
H: phi x
----------------------------------------
(exists (x:'a),
phi x && try find x:'a such that phi x in phi x else false = phi x)
=> try find x:'a such that phi x in phi x else false

[> Line 514: (auto) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:'a -> bool,x:'a
H: phi x
----------------------------------------
(forall (x:'a), not (phi x)) &&
try find x:'a such that phi x in phi x else false = false =>
try find x:'a such that phi x in phi x else false

[> Line 515: ((intro);by (have)) [goal> lemma choose_spec is proved

lemma choose_spec {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (phi:'a -> bool,x:'a), phi x => phi (choose phi)
Exiting proof mode.

Goal try_choose :
phi x => try find x:'a such that phi x in f x else g = f (choose phi)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: f:'a -> 'b,g:'b,phi:'a -> bool,x:'a
----------------------------------------
phi x => try find x:'a such that phi x in f x else g = f (choose phi)

[> Line 524: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: f:'a -> 'b,g:'b,phi:'a -> bool,x:'a
H: phi x
----------------------------------------
try find x:'a such that phi x in f x else g = f (choose phi)

[> Line 525: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: f:'a -> 'b,g:'b,phi:'a -> bool,x:'a
H: phi x
----------------------------------------
if (exists (x:'a), phi x) then f (choose phi) else g = f (choose phi)

[> Line 526: ((rewrite);(intro)) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: f:'a -> 'b,g:'b,phi:'a -> bool,x:'a
H: phi x
----------------------------------------
exists (x:'a), phi x

[> Line 527: by (exists) [goal> lemma try_choose is proved

lemma try_choose {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
forall (phi:'a -> bool,f:'a -> 'b,g:'b,x:'a),
phi x => try find x:'a such that phi x in f x else g = f (choose phi)
Exiting proof mode.

Goal forall_exists :
(forall (x:'a), exists (y:'b), phi x y) =
exists (y':'a -> 'b), forall (x:'a), phi x (y' x)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool
----------------------------------------
(forall (x:'a), exists (y:'b), phi x y) =
exists (y':'a -> 'b), forall (x:'a), phi x (y' x)

[> Line 536: ((rewrite);(split)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool
----------------------------------------
(forall (x:'a), exists (y:'b), phi x y) =>
exists (y':'a -> 'b), forall (x:'a), phi x (y' x)

[> Line 537: (intro) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool
H: forall (x:'a), exists (y:'b), phi x y
----------------------------------------
exists (y':'a -> 'b), forall (x:'a), phi x (y' x)

[> Line 538: (exists) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool
H: forall (x:'a), exists (y:'b), phi x y
----------------------------------------
forall (x:'a), phi x ((fun (x:'a) => choose (fun (y:'b) => phi x y)) x)

[> Line 539: ((intro);(simpl)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool,x:'a
H: forall (x:'a), exists (y:'b), phi x y
----------------------------------------
phi x (choose (fun (y:'b) => phi x y))

[> Line 540: (have) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool,x:'a,y:'b
H: forall (x:'a), exists (y:'b), phi x y
Hy: phi x y
----------------------------------------
phi x (choose (fun (y:'b) => phi x y))

[> Line 544: ((have); 1: by (auto)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool,x:'a,y:'b
H: forall (x:'a), exists (y:'b), phi x y
Hy: phi x y
----------------------------------------
(fun (y:'b) => phi x y) (choose (fun (y:'b) => phi x y))

[> Line 545: (apply) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool,x:'a,y:'b
H: forall (x:'a), exists (y:'b), phi x y
Hy: phi x y
----------------------------------------
(fun (y:'b) => phi x y) y

[> Line 546: ((simpl);(assumption)) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool
----------------------------------------
(exists (y':'a -> 'b), forall (x:'a), phi x (y' x)) =>
forall (x:'a), exists (y:'b), phi x y

[> Line 547: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool,x:'a,y':'a -> 'b
H: forall (x:'a), phi x (y' x)
----------------------------------------
exists (y:'b), phi x y

[> Line 548: (exists) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a, 'b
Variables: phi:'a -> 'b -> bool,x:'a,y':'a -> 'b
H: forall (x:'a), phi x (y' x)
----------------------------------------
phi x (y' x)

[> Line 549: by (apply) [goal> lemma forall_exists is proved

lemma forall_exists {'P:system} @system:(set:'P; equiv:None) ['a 'b] :
forall (phi:'a -> 'b -> bool),
(forall (x:'a), exists (y:'b), phi x y) =
exists (y':'a -> 'b), forall (x:'a), phi x (y' x)
Exiting proof mode.

Goal implies_exists :
(phi => exists (j:'a), psi j) = exists (x:'a), phi => psi x
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool,psi:'a -> bool
----------------------------------------
(phi => exists (j:'a), psi j) = exists (x:'a), phi => psi x

[> Line 556: ((rewrite);(split)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool[const],psi:'a -> bool
----------------------------------------
(phi => exists (j:'a), psi j) => exists (x:'a), phi => psi x

[> Line 557: (intro) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool[const],psi:'a -> bool
H: phi => exists (j:'a), psi j
----------------------------------------
exists (x:'a), phi => psi x

[> Line 558: (case) [goal> Focused goal (1/3):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool[const],psi:'a -> bool
H: phi => exists (j:'a), psi j
----------------------------------------
phi => exists (x:'a), true => psi x

[> Line 559: (intro) [goal> Focused goal (1/3):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool[const],psi:'a -> bool
H: phi => exists (j:'a), psi j
phi: phi
----------------------------------------
exists (x:'a), true => psi x

[> Line 560: ((have); 1: by (apply)) [goal> Focused goal (1/3):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool[const],psi:'a -> bool,x:'a
H: phi => exists (j:'a), psi j
_: psi x
phi: phi
----------------------------------------
exists (x:'a), true => psi x

[> Line 561: by (exists) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool[const],psi:'a -> bool
H: phi => exists (j:'a), psi j
----------------------------------------
not phi => exists (x:'a), false => psi x

[> Line 562: (intro) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool[const],psi:'a -> bool
H: phi => exists (j:'a), psi j
_: not phi
----------------------------------------
exists (x:'a), false => psi x

[> Line 563: by (exists) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool[const],psi:'a -> bool
----------------------------------------
(exists (x:'a), phi => psi x) => phi => exists (j:'a), psi j

[> Line 564: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: phi:bool[const],psi:'a -> bool,x:'a
H: phi => psi x
H': phi
----------------------------------------
exists (j:'a), psi j

[> Line 565: by (exists) [goal> lemma implies_exists is proved

lemma implies_exists {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (phi:bool,psi:'a -> bool),
(phi => exists (j:'a), psi j) = exists (x:'a), phi => psi x
Exiting proof mode.

axiom le_trans {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y,z:'a), x <= y => y <= z => x <= z
axiom lt_trans {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y,z:'a), x < y => y < z => x < z
axiom lt_le_trans {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y,z:'a), x < y => y <= z => x < z
axiom le_lt_trans {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y,z:'a), x <= y => y < z => x < z
axiom lt_charac {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), x < y <=> x <> y && x <= y
axiom le_not_lt_impl_eq {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), x <= y => not (x < y) => x = y
Goal lt_impl_le :
x < y => x <= y
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x,y:'a
----------------------------------------
x < y => x <= y

[> Line 580: by (rewrite) [goal> lemma lt_impl_le is proved

lemma lt_impl_le {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), x < y => x <= y
Exiting proof mode.

Goal not_lt_refl :
not (x < x)
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x:'a
----------------------------------------
not (x < x)

[> Line 583: (auto) [goal> lemma not_lt_refl is proved

lemma not_lt_refl {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x:'a), not (x < x)
Exiting proof mode.

Goal lt_irrefl :
x < x <=> false
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Type variables: 'a
Variables: x:'a
----------------------------------------
x < x <=> false

[> Line 586: (auto) [goal> lemma lt_irrefl is proved

lemma lt_irrefl {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x:'a), x < x <=> false
Exiting proof mode.

axiom le_impl_eq_lt {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,y:'a), x <= y => x = y || x < y
axiom le_refl_index {'P:system} @system:(set:'P; equiv:None) :
forall (x:index), x <= x
Goal le_refl_index_eq :
x <= x = true
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: x:index
----------------------------------------
x <= x = true

[> Line 599: by (rewrite) [goal> lemma le_refl_index_eq is proved

lemma le_refl_index_eq {'P:system} @system:(set:'P; equiv:None) :
forall (x:index), x <= x = true
Exiting proof mode.

Goal le_pred_lt :
t <= pred t' = t < t'
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: t,t':timestamp
----------------------------------------
t <= pred t' = t < t'

[> Line 603: by (rewrite) [goal> lemma le_pred_lt is proved

lemma le_pred_lt {'P:system} @system:(set:'P; equiv:None) :
forall (t,t':timestamp), t <= pred t' = t < t'
Exiting proof mode.

Goal neq_le_pred_le :
t <> t' => t <= t' = t <= pred t'
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: t,t':timestamp
----------------------------------------
t <> t' => t <= t' = t <= pred t'

[> Line 607: by (rewrite) [goal> lemma neq_le_pred_le is proved

lemma neq_le_pred_le {'P:system} @system:(set:'P; equiv:None) :
forall (t,t':timestamp), t <> t' => t <= t' = t <= pred t'
Exiting proof mode.

axiom le_lt {'P:system} @system:(set:'P; equiv:None) ['a] :
forall (x,x':'a), x <> x' => x <= x' = x < x'
[warning>Loaded "Logic.sp".
<]axiom empty_set_is_empty {'P:system} @system:(set:'P; equiv:None) :
forall (x:message), not (mem x empty_set)
[warning>Loaded "Set.sp".
<]new predicate:
predicate (|>) {set:system} ['a 'b] {set: u:'a,m:'b} =
Exists (f:'a -> 'b[adv, glob]), [f u = m]
new predicate:
predicate (|1>) {set:system} ['a 'b 'c] {set: u:'a -> 'b,m:'a -> 'c} =
Exists (f:'b -> 'c[adv, glob]), [forall (x:'a), f (u x) = m x]
[warning>Loaded "DeductionSyntax.sp".
<]Goal unnamed1 :
forall u:'b[glob],v:'a -> 'c[glob],
Let u0 = fun (_x:'a) => u in
$(u0 |1>{Empty} fun (x:'a) => v x) -> $(u |>{Empty} fun (x:'a) => v x)
[goal> Focused goal (1/1):
Systems: Empty
Type variables: 'a, 'b, 'c
Variables: u:'b[glob],v:'a -> 'c[glob]
----------------------------------------
Let u0 = fun (_x:'a) => u in
$(u0 |1> fun (x:'a) => v x) -> $(u |> fun (x:'a) => v x)

[> Line 11: (intro) [goal> Focused goal (1/1):
Systems: Empty
Type variables: 'a, 'b, 'c
Variables: u:'b[glob],v:'a -> 'c[glob]
H: $(u0 |1> fun (x:'a) => v x)
u0 := fun (_x:'a) => u
----------------------------------------
$(u |> fun (x:'a) => v x)

[> Line 12: (rewrite) [goal> Focused goal (1/1):
Systems: Empty
Type variables: 'a, 'b, 'c
Variables: u:'b[glob],v:'a -> 'c[glob]
H: $(u0 |1> fun (x:'a) => v x)
u0 := fun (_x:'a) => u
----------------------------------------
Exists (f:'b -> 'a -> 'c[adv, glob]), [f u = (fun (x:'a) => v x)]

[> Line 13: (rewrite) [goal> Focused goal (1/1):
Systems: Empty
Type variables: 'a, 'b, 'c
Variables: u:'b[glob],v:'a -> 'c[glob]
H: Exists (f:'b -> 'c[adv, glob]),
[forall (x:'a), f (u0 x) = (fun (x:'a) => v x) x]
u0 := fun (_x:'a) => u
----------------------------------------
Exists (f:'b -> 'a -> 'c[adv, glob]), [f u = (fun (x:'a) => v x)]

[> Line 14: (destruct) [goal> Focused goal (1/1):
Systems: Empty
Type variables: 'a, 'b, 'c
Variables: f:'b -> 'c[adv, glob],u:'b[glob],v:'a -> 'c[glob]
H: [forall (x:'a), f (u0 x) = (fun (x:'a) => v x) x]
u0 := fun (_x:'a) => u
----------------------------------------
Exists (f:'b -> 'a -> 'c[adv, glob]), [f u = (fun (x:'a) => v x)]

[> Line 14: ((exists);(intro)) [goal> Focused goal (1/1):
System: Empty
Type variables: 'a, 'b, 'c
Variables: f:'b -> 'c[adv, glob],u:'b[glob],v:'a -> 'c[glob]
H: [forall (x:'a), f (u0 x) = v x]
u0 := fun (_x:'a) => u
----------------------------------------
(fun (x:'a) => f u) = (fun (x:'a) => v x)

[> Line 16: ((apply);(intro)) [goal> Focused goal (1/1):
System: Empty
Type variables: 'a, 'b, 'c
Variables: f:'b -> 'c[adv, glob],u:'b[glob],v:'a -> 'c[glob],x:'a
H: [forall (x:'a), f (u0 x) = v x]
u0 := fun (_x:'a) => u
----------------------------------------
f u = v x

[> Line 17: (rewrite) [goal> Focused goal (1/1):
System: Empty
Type variables: 'a, 'b, 'c
Variables: f:'b -> 'c[adv, glob],u:'b[glob],v:'a -> 'c[glob],x:'a
H: [forall (x:'a), f u = v x]
u0 := fun (_x:'a) => u
----------------------------------------
f u = v x

[> Line 18: (apply) [goal> lemma unnamed1 is proved

global lemma unnamed1 @system:Empty ['a 'b 'c] :
Forall (u:'b[glob],v:'a -> 'c[glob]),
Let u0 = fun (_x:'a) => u in
$(u0 |1>{Empty} fun (x:'a) => v x) -> $(u |>{Empty} fun (x:'a) => v x)
Exiting proof mode.

global axiom frame_from_frame {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => frame@t |1>{P}
fun (t,t':timestamp) => if (t' <= t) then frame@t')
New deduction hint frame_from_frame :
∀{P:system} , @system:(P), ∀ x ⊢ frame@x ▷ λ t' ⇒ (frame@t' |
t' <= x)
global axiom exec_from_frame {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => frame@t |1>{P}
fun (t,t':timestamp) => if (t' <= t) then exec@t' else witness)
New deduction hint exec_from_frame :
∀{P:system} , @system:(P), ∀ x ⊢ frame@x ▷ λ t' ⇒ (exec@t' |
t' <= x)
global axiom output_from_frame {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => frame@t |1>{P}
fun (t,t':timestamp) => if (t' <= t && exec@t') then output@t')
New deduction hint output_from_frame :
∀{P:system} , @system:(P), ∀ x ⊢ frame@x ▷ λ t' ⇒ (output@t' |
t' <= x && exec@t')
global axiom input_from_frame {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => frame@t |1>{P}
fun (t,t':timestamp) => if (pred t' <= t) then input@t')
New deduction hint input_from_frame :
∀{P:system} , @system:(P), ∀ x ⊢ frame@x ▷ λ t' ⇒ (input@t' |
pred t' <= x)
global axiom exec_from_frame {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => frame@t |1>{P}
fun (t,t':timestamp) => if (t' <= t) then exec@t' else witness)
New deduction hint exec_from_frame :
∀{P:system} , @system:(P), ∀ x ⊢ frame@x ▷ λ t' ⇒ (exec@t' |
t' <= x)
global axiom output_from_frame {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => frame@t |1>{P}
fun (t,t':timestamp) => if (t' <= t && exec@t') then output@t')
New deduction hint output_from_frame :
∀{P:system} , @system:(P), ∀ x ⊢ frame@x ▷ λ t' ⇒ (output@t' |
t' <= x && exec@t')
global axiom input_from_frame {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => frame@t |1>{P}
fun (t,t':timestamp) => if (pred t' <= t) then input@t')
New deduction hint input_from_frame :
∀{P:system} , @system:(P), ∀ x ⊢ frame@x ▷ λ t' ⇒ (input@t' |
pred t' <= x)
global axiom transcript_from_frame {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => frame@t |1>{P}
fun (t,t':timestamp) => if (t' <= t) then transcript@t')
New deduction hint transcript_from_frame :
∀{P:system} , @system:(P), ∀ x ⊢ frame@x ▷ λ t' ⇒ (transcript@t' |
t' <= x)
global axiom transcript_from_transcript {P:system}
@system:(set:P; equiv:None) :
$(fun (t:timestamp) => transcript@t |1>{P}
fun (t,t':timestamp) => if (t' <= t) then transcript@t')
New deduction hint transcript_from_transcript :
∀{P:system} , @system:(P), ∀ x ⊢ transcript@x ▷ λ t' ⇒ (
transcript@t' |
t' <= x)
global axiom exec_from_transcript {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => transcript@t |1>{P}
fun (t,t':timestamp) => if (t' <= t) then exec@t' else witness)
New deduction hint exec_from_transcript :
∀{P:system} , @system:(P), ∀ x ⊢ transcript@x ▷ λ t' ⇒ (
exec@t' | t' <= x)
global axiom output_from_transcript {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => transcript@t |1>{P}
fun (t,t':timestamp) => if (t' <= t && exec@t') then output@t')
New deduction hint output_from_transcript :
∀{P:system} , @system:(P), ∀ x ⊢ transcript@x ▷ λ t' ⇒ (
output@t' |
t' <= x && exec@t')
global axiom input_from_transcript {P:system} @system:(set:P; equiv:None) :
$(fun (t:timestamp) => transcript@t |1>{P}
fun (t,t':timestamp) => if (pred t' <= t) then input@t')
New deduction hint input_from_transcript :
∀{P:system} , @system:(P), ∀ x ⊢ transcript@x ▷ λ t' ⇒ (
input@t' |
pred t' <= x)
[warning>Loaded "Deduction.sp".
<]axiom exec_not_init {'P:system} @system:(set:'P; equiv:None) :
forall (tau:timestamp),
init < tau => exec@tau = (exec@pred tau && cond@tau)
axiom exec_init {'P:system} @system:(set:'P; equiv:None) :
forall (tau:timestamp), tau = init => exec@tau = true
axiom cond_init {'P:system} @system:(set:'P; equiv:None) :
forall (tau:timestamp), tau = init => cond@tau = true
Goal exec_le :
tau' <= tau => exec@tau => exec@tau'
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau,tau':timestamp
----------------------------------------
tau' <= tau => exec@tau => exec@tau'

[> Line 17: ((induction);(intro)) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau,tau':timestamp[const]
Hexec: exec@tau
Hle: tau' <= tau
IH: forall (tau0:timestamp),
tau0 < tau => tau' <= tau0 => exec@tau0 => exec@tau'
----------------------------------------
exec@tau'

[> Line 18: (case) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau,tau':timestamp[const]
Hexec: exec@tau
Hle: tau' <= tau
IH: forall (tau0:timestamp),
tau0 < tau => tau' <= tau0 => exec@tau0 => exec@tau'
----------------------------------------
tau = tau' => exec@tau'

[> Line 19: (auto) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau,tau':timestamp[const]
Hexec: exec@tau
Hle: tau' <= tau
IH: forall (tau0:timestamp),
tau0 < tau => tau' <= tau0 => exec@tau0 => exec@tau'
----------------------------------------
not (tau = tau') => exec@tau'

[> Line 20: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau,tau':timestamp[const]
Hexec: exec@tau
Hle: tau' <= tau
Hneq: not (tau = tau')
IH: forall (tau0:timestamp),
tau0 < tau => tau' <= tau0 => exec@tau0 => exec@tau'
----------------------------------------
exec@tau'

[> Line 21: (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau,tau':timestamp[const]
Hexec: exec@pred tau && cond@tau
Hle: tau' <= tau
Hneq: not (tau = tau')
IH: forall (tau0:timestamp),
tau0 < tau => tau' <= tau0 => exec@tau0 => exec@tau'
----------------------------------------
exec@tau'

[> Line 22: by (apply) [goal> lemma exec_le is proved

lemma exec_le {'P:system} @system:(set:'P; equiv:None) :
forall (tau,tau':timestamp), tau' <= tau => exec@tau => exec@tau'
Exiting proof mode.

Goal exec_cond :
happens(tau) => exec@tau => cond@tau
[goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau:timestamp
----------------------------------------
happens(tau) => exec@tau => cond@tau

[> Line 27: (intro) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau:timestamp[const]
Hap: happens(tau)
Hexec: exec@tau
----------------------------------------
cond@tau

[> Line 28: ((case);(intro)) [goal> Focused goal (1/2):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau:timestamp[const]
Hap: happens(tau)
Hexec: exec@tau
_: init < tau
----------------------------------------
cond@tau

[> Line 29: by (rewrite) [goal> Focused goal (1/1):
System variables: 'P
System: (set:'P; equiv:None)
Variables: tau:timestamp[const]
Hap: happens(tau)
Hexec: exec@tau
_: not (init < tau)
----------------------------------------
cond@tau

[> Line 30: by (rewrite) [goal> lemma exec_cond is proved

lemma exec_cond {'P:system} @system:(set:'P; equiv:None) :
forall (tau:timestamp), happens(tau) => exec@tau => cond@tau
Exiting proof mode.

axiom executability {'P:system} @system:(set:'P; equiv:None) :
forall (t:timestamp),
happens(t) => exec@t => forall (t0:timestamp), t0 <= t => exec@t0
[warning>Loaded "Classic.sp".
<][warning>Loaded "Core.sp".
<]


(* HELPING LEMMAS *)

(* Last update lemmas: basic reasoning about the memory cells.
Each lemma has two versions, one for the tag and one for the reader.
We describe below the tag's lemmas, the reader's lemmas are similar. *)

(* lastupdate_pure_tag
This lemmas reasons on the order of actions in the execution trace (i.e. pure
trace formulas). It states that a timestamp tau either happens before any
action T(i,j), or there exists an action T(i,j) such that T(i,j) is the last
action of the form T(i,_) that happens before tau (i.e. any other T(i,j') that
happens before tau actually happens before T(i,j)). *)

(* lastupdate_init_tag
This lemma states that, for any timestamp tau, if no tag T(i,j) has yet played,
then sT(i)@tau = sT(i)@init. *)

(* lastupdate_T
This lemma states that, for any timestamp tau, if T(i,j) is the last action of
the form T(i,_) that happens before tau, then sT(i)@tau = sT(i)@T(i,j). *)

(* lastupdateTag
This lemma brings together all previous lemmas. For any timestamp tau:
- either no tag T(i,j) has played before, in that case sT(i)@tau = sT(i)@init;
- or there exists an action T(i,j) such that T(i,j) is the last action of the
form T(i,_) that happens before tau, in that case sT(i)@tau = sT(i)@T(i,j). *)

lemma lastupdate_pure_tag (i:index,tau:timestamp):
happens(tau) => (
(forall j, happens(T(i,j)) => T(i,j)>tau) ||
(exists j, happens(T(i,j)) && T(i,j)<=tau &&
forall j', happens(T(i,j')) && T(i,j')<=tau => T(i,j')<=T(i,j))).
Goal lastupdate_pure_tag :
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i:index,tau:timestamp
----------------------------------------
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)



induction tau => tau IH Hap.
[> Line 94: ((induction);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)



case tau.
[> Line 95: (case) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
----------------------------------------
tau = init =>
(forall (j:index), happens(T(i, j)) => T(i, j) > init) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= init &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= init => T(i, j') <= T(i, j)




(* init *)
+ by intro _; left.
[> Line 98: by ((intro);(left)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
----------------------------------------
(exists (i0,j:index), tau = T(i0, j)) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)




(* T(i0,j) *)
+ intro [i0 j Eq]; subst tau, T(i0,j).
[> Line 101: ((intro);(subst)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const]
Hap: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) ||
exists (j0:index),
happens(T(i, j0)) &&
T(i, j0) <= T(i0, j) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j) => T(i, j') <= T(i, j0)



(* 1st case: i<>i0 *)
case (i<>i0) => //.
[> Line 103: ((case);(intro)) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const]
Hap: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
i <> i0 =>
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) ||
exists (j0:index),
happens(T(i, j0)) &&
T(i, j0) <= T(i0, j) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j) => T(i, j') <= T(i, j0)



intro Neq.
[> Line 104: (intro) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const]
Hap: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Neq: i <> i0
----------------------------------------
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) ||
exists (j0:index),
happens(T(i, j0)) &&
T(i, j0) <= T(i0, j) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j) => T(i, j') <= T(i, j0)



use IH with pred(T(i0,j)) => //.
[> Line 105: ((have);(intro)) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const]
H: (forall (j0:index), happens(T(i, j0)) => T(i, j0) > pred (T(i0, j)))
||
exists (j0:index),
happens(T(i, j0)) &&
T(i, j0) <= pred (T(i0, j)) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (T(i0, j)) =>
T(i, j') <= T(i, j0)
Hap: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Neq: i <> i0
----------------------------------------
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) ||
exists (j0:index),
happens(T(i, j0)) &&
T(i, j0) <= T(i0, j) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j) => T(i, j') <= T(i, j0)



destruct H as [H1 | [j0 H2]].
[> Line 106: (destruct) [goal> Focused goal (1/5):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const]
H1: forall (j0:index), happens(T(i, j0)) => T(i, j0) > pred (T(i0, j))
Hap: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Neq: i <> i0
----------------------------------------
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) ||
exists (j0:index),
happens(T(i, j0)) &&
T(i, j0) <= T(i0, j) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j) => T(i, j') <= T(i, j0)



- left; intro j0 HapT; by use H1 with j0.
[> Line 107: ((left);((intro);by (have))) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const]
H2: happens(T(i, j0)) &&
T(i, j0) <= pred (T(i0, j)) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (T(i0, j)) =>
T(i, j') <= T(i, j0)
Hap: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Neq: i <> i0
----------------------------------------
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) ||
exists (j0:index),
happens(T(i, j0)) &&
T(i, j0) <= T(i0, j) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j) => T(i, j') <= T(i, j0)



- right; destruct H2 as [_ _ H2]; exists j0.
[> Line 108: ((right);((destruct);(exists))) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const]
H2: forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (T(i0, j)) =>
T(i, j') <= T(i, j0)
Hap: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Neq: i <> i0
_: T(i, j0) <= pred (T(i0, j))
_: happens(T(i, j0))
----------------------------------------
happens(T(i, j0)) &&
T(i, j0) <= T(i0, j) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j) => T(i, j') <= T(i, j0)



intro /= _ _.
[> Line 109: (intro) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j,j',j0:index[const]
H2: forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (T(i0, j)) =>
T(i, j') <= T(i, j0)
Hap: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Neq: i <> i0
_: happens(T(i, j')) && T(i, j') <= T(i0, j)
_: T(i, j0) <= pred (T(i0, j))
_: happens(T(i, j0))
----------------------------------------
T(i, j') <= T(i, j0)


by apply H2.
[> Line 110: by (apply) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const]
Hap: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
not (i <> i0) =>
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) ||
exists (j0:index),
happens(T(i, j0)) &&
T(i, j0) <= T(i0, j) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j) => T(i, j') <= T(i, j0)




(* 2nd case: i<>i0 *)
intro _; subst i0, i.
[> Line 113: ((intro);(subst)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,j:index[const]
Hap: happens(T(i, j))
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
_: not (i <> i)
----------------------------------------
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i, j)) ||
exists (j0:index),
happens(T(i, j0)) &&
T(i, j0) <= T(i, j) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i, j) => T(i, j') <= T(i, j0)



by right; exists j.
[> Line 114: by ((right);(exists)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
----------------------------------------
(exists (jj,ii:index), tau = R(jj, ii)) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)




(* R(jj,ii) *)
+ intro [jj ii Eq]; subst tau, R(jj,ii).
[> Line 117: ((intro);(subst)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,jj:index[const]
Hap: happens(R(jj, ii))
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R(jj, ii)) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R(jj, ii) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)



use IH with pred(R(jj,ii)) => //.
[> Line 118: ((have);(intro)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,jj:index[const]
H: (forall (j:index), happens(T(i, j)) => T(i, j) > pred (R(jj, ii)))
||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= pred (R(jj, ii)) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R(jj, ii)) =>
T(i, j') <= T(i, j)
Hap: happens(R(jj, ii))
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R(jj, ii)) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R(jj, ii) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)



destruct H as [H1 | [j H2]].
[> Line 119: (destruct) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,jj:index[const]
H1: forall (j:index), happens(T(i, j)) => T(i, j) > pred (R(jj, ii))
Hap: happens(R(jj, ii))
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R(jj, ii)) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R(jj, ii) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)



- left; intro j HapT; by use H1 with j.
[> Line 120: ((left);((intro);by (have))) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
H2: happens(T(i, j)) &&
T(i, j) <= pred (R(jj, ii)) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R(jj, ii)) =>
T(i, j') <= T(i, j)
Hap: happens(R(jj, ii))
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R(jj, ii)) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R(jj, ii) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)



- right; destruct H2 as [_ _ H2].
[> Line 121: ((right);(destruct)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
H2: forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R(jj, ii)) =>
T(i, j') <= T(i, j)
Hap: happens(R(jj, ii))
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
_: T(i, j) <= pred (R(jj, ii))
_: happens(T(i, j))
----------------------------------------
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R(jj, ii) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)



exists j.
[> Line 122: (exists) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
H2: forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R(jj, ii)) =>
T(i, j') <= T(i, j)
Hap: happens(R(jj, ii))
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
_: T(i, j) <= pred (R(jj, ii))
_: happens(T(i, j))
----------------------------------------
happens(T(i, j)) &&
T(i, j) <= R(jj, ii) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)



intro /= _ _.
[> Line 123: (intro) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,j,j',jj:index[const]
H2: forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R(jj, ii)) =>
T(i, j') <= T(i, j)
Hap: happens(R(jj, ii))
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
_: happens(T(i, j')) && T(i, j') <= R(jj, ii)
_: T(i, j) <= pred (R(jj, ii))
_: happens(T(i, j))
----------------------------------------
T(i, j') <= T(i, j)


by apply H2.
[> Line 124: by (apply) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
----------------------------------------
(exists (jj:index), tau = R1(jj)) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)




(* R1(jj) *)
+ intro [jj Eq]; subst tau, R1(jj).
[> Line 127: ((intro);(subst)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,jj:index[const]
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R1(jj)) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R1(jj) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)



use IH with pred(R1(jj)) => //.
[> Line 128: ((have);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,jj:index[const]
H: (forall (j:index), happens(T(i, j)) => T(i, j) > pred (R1(jj))) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= pred (R1(jj)) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R1(jj)) => T(i, j') <= T(i, j)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R1(jj)) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R1(jj) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)



destruct H as [H1 | [j H2]].
[> Line 129: (destruct) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,jj:index[const]
H1: forall (j:index), happens(T(i, j)) => T(i, j) > pred (R1(jj))
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R1(jj)) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R1(jj) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)



- left; intro j HapT; by use H1 with j.
[> Line 130: ((left);((intro);by (have))) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j,jj:index[const]
H2: happens(T(i, j)) &&
T(i, j) <= pred (R1(jj)) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R1(jj)) => T(i, j') <= T(i, j)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R1(jj)) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R1(jj) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)



- right; destruct H2 as [_ _ H2].
[> Line 131: ((right);(destruct)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j,jj:index[const]
H2: forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R1(jj)) => T(i, j') <= T(i, j)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
_: T(i, j) <= pred (R1(jj))
_: happens(T(i, j))
----------------------------------------
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= R1(jj) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)



exists j.
[> Line 132: (exists) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j,jj:index[const]
H2: forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R1(jj)) => T(i, j') <= T(i, j)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
_: T(i, j) <= pred (R1(jj))
_: happens(T(i, j))
----------------------------------------
happens(T(i, j)) &&
T(i, j) <= R1(jj) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)



intro /= _ _.
[> Line 133: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j,j',jj:index[const]
H2: forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R1(jj)) => T(i, j') <= T(i, j)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
_: happens(T(i, j')) && T(i, j') <= R1(jj)
_: T(i, j) <= pred (R1(jj))
_: happens(T(i, j))
----------------------------------------
T(i, j') <= T(i, j)


by apply H2.
[> Line 134: by (apply) [goal> lemma lastupdate_pure_tag is proved


Qed.
lemma lastupdate_pure_tag @system:(set:default; equiv:None) :
forall (i:index,tau:timestamp),
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) ||
exists (j:index),
happens(T(i, j)) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Exiting proof mode.




lemma lastupdate_pure_reader (ii:index,tau:timestamp):
happens(tau) => (
(forall jj, happens(R(jj,ii)) => R(jj,ii)>tau) ||
(exists jj, happens(R(jj,ii)) && R(jj,ii)<=tau &&
forall jj',
happens(R(jj',ii)) && R(jj',ii)<=tau => R(jj',ii)<=R(jj,ii))).
Goal lastupdate_pure_reader :
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii:index,tau:timestamp
----------------------------------------
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)



induction tau => tau IH Hap.
[> Line 145: ((induction);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 =>
R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)



case tau.
[> Line 146: (case) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 =>
R(jj', ii) <= R(jj, ii)
----------------------------------------
tau = init =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > init) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= init &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= init => R(jj', ii) <= R(jj, ii)




(* init *)
+ by intro _; left.
[> Line 149: by ((intro);(left)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 =>
R(jj', ii) <= R(jj, ii)
----------------------------------------
(exists (i,j:index), tau = T(i, j)) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)




(* T(i,j) *)
+ intro [i j Eq]; subst tau, T(i,j).
[> Line 152: ((intro);(subst)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j:index[const]
Hap: happens(T(i, j))
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > T(i, j)) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= T(i, j) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) => R(jj', ii) <= R(jj, ii)



use IH with pred(T(i,j)) => //.
[> Line 153: ((have);(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j:index[const]
H: (forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > pred (T(i, j)))
||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= pred (T(i, j)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (T(i, j)) =>
R(jj', ii) <= R(jj, ii)
Hap: happens(T(i, j))
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > T(i, j)) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= T(i, j) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) => R(jj', ii) <= R(jj, ii)



destruct H as [H1 | [jj H2]].
[> Line 154: (destruct) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,ii,j:index[const]
H1: forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > pred (T(i, j))
Hap: happens(T(i, j))
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > T(i, j)) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= T(i, j) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) => R(jj', ii) <= R(jj, ii)



- left; intro jj HapR; by use H1 with jj => //.
[> Line 155: (((left);((intro);by (have)));(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
H2: happens(R(jj, ii)) &&
R(jj, ii) <= pred (T(i, j)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (T(i, j)) =>
R(jj', ii) <= R(jj, ii)
Hap: happens(T(i, j))
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > T(i, j)) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= T(i, j) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) => R(jj', ii) <= R(jj, ii)



- right; destruct H2 as [_ _ H2].
[> Line 156: ((right);(destruct)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (T(i, j)) =>
R(jj', ii) <= R(jj, ii)
Hap: happens(T(i, j))
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
_: R(jj, ii) <= pred (T(i, j))
_: happens(R(jj, ii))
----------------------------------------
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= T(i, j) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) => R(jj', ii) <= R(jj, ii)



exists jj.
[> Line 157: (exists) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (T(i, j)) =>
R(jj', ii) <= R(jj, ii)
Hap: happens(T(i, j))
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
_: R(jj, ii) <= pred (T(i, j))
_: happens(R(jj, ii))
----------------------------------------
happens(R(jj, ii)) &&
R(jj, ii) <= T(i, j) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) => R(jj', ii) <= R(jj, ii)



intro /= _ _.
[> Line 158: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j,jj,jj':index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (T(i, j)) =>
R(jj', ii) <= R(jj, ii)
Hap: happens(T(i, j))
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
_: happens(R(jj', ii)) && R(jj', ii) <= T(i, j)
_: R(jj, ii) <= pred (T(i, j))
_: happens(R(jj, ii))
----------------------------------------
R(jj', ii) <= R(jj, ii)


by apply H2.
[> Line 159: by (apply) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 =>
R(jj', ii) <= R(jj, ii)
----------------------------------------
(exists (jj,ii0:index), tau = R(jj, ii0)) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)




(* R(jj,ii0) *)
+ intro [jj ii0 Eq]; subst tau, R(jj,ii0).
[> Line 162: ((intro);(subst)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const]
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R(jj, ii0) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) =>
R(jj', ii) <= R(jj0, ii)



case (ii<>ii0) => //.
[> Line 163: ((case);(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const]
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
ii <> ii0 =>
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R(jj, ii0) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) =>
R(jj', ii) <= R(jj0, ii)



(* 1st case: ii<>ii0 *)
- intro Neq.
[> Line 165: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const]
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Neq: ii <> ii0
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R(jj, ii0) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) =>
R(jj', ii) <= R(jj0, ii)



use IH with pred(R(jj,ii0)) => //.
[> Line 166: ((have);(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const]
H: (forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > pred (R(jj, ii0)))
||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= pred (R(jj, ii0)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R(jj, ii0)) =>
R(jj', ii) <= R(jj0, ii)
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Neq: ii <> ii0
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R(jj, ii0) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) =>
R(jj', ii) <= R(jj0, ii)



destruct H as [H1 | [jj0 H2]].
[> Line 167: (destruct) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const]
H1: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > pred (R(jj, ii0))
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Neq: ii <> ii0
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R(jj, ii0) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) =>
R(jj', ii) <= R(jj0, ii)



left; intro jj0 HapR; by use H1 with jj0 => //.
[> Line 168: (((left);((intro);by (have)));(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const]
H2: happens(R(jj0, ii)) &&
R(jj0, ii) <= pred (R(jj, ii0)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R(jj, ii0)) =>
R(jj', ii) <= R(jj0, ii)
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Neq: ii <> ii0
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R(jj, ii0) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) =>
R(jj', ii) <= R(jj0, ii)



right; destruct H2 as [_ _ H2]; exists jj0.
[> Line 169: ((right);((destruct);(exists))) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R(jj, ii0)) =>
R(jj', ii) <= R(jj0, ii)
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Neq: ii <> ii0
_: R(jj0, ii) <= pred (R(jj, ii0))
_: happens(R(jj0, ii))
----------------------------------------
happens(R(jj0, ii)) &&
R(jj0, ii) <= R(jj, ii0) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) => R(jj', ii) <= R(jj0, ii)



simpl.
[> Line 170: (simpl) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R(jj, ii0)) =>
R(jj', ii) <= R(jj0, ii)
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Neq: ii <> ii0
_: R(jj0, ii) <= pred (R(jj, ii0))
_: happens(R(jj0, ii))
----------------------------------------
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) => R(jj', ii) <= R(jj0, ii)



intro jj'.
[> Line 171: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj',jj0:index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R(jj, ii0)) =>
R(jj', ii) <= R(jj0, ii)
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Neq: ii <> ii0
_: R(jj0, ii) <= pred (R(jj, ii0))
_: happens(R(jj0, ii))
----------------------------------------
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) => R(jj', ii) <= R(jj0, ii)


intro Hyp.
[> Line 172: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj',jj0:index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R(jj, ii0)) =>
R(jj', ii) <= R(jj0, ii)
Hap: happens(R(jj, ii0))
Hyp: happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0)
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Neq: ii <> ii0
_: R(jj0, ii) <= pred (R(jj, ii0))
_: happens(R(jj0, ii))
----------------------------------------
R(jj', ii) <= R(jj0, ii)


by apply H2.
[> Line 173: by (apply) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const]
Hap: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
not (ii <> ii0) =>
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R(jj, ii0) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii0) =>
R(jj', ii) <= R(jj0, ii)




(* 2nd case: ii<>ii0 *)
- intro Eq; subst ii0, ii.
[> Line 176: ((intro);(subst)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,jj:index[const]
Eq: not (ii <> ii)
Hap: happens(R(jj, ii))
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii)) ||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R(jj, ii) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj, ii) =>
R(jj', ii) <= R(jj0, ii)



right; exists jj; split => //.
[> Line 177: (((right);((exists);(split)));(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Hap: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 =>
R(jj', ii) <= R(jj, ii)
----------------------------------------
(exists (jj:index), tau = R1(jj)) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)




(* R1(jj) *)
+ intro [jj Eq]; subst tau, R1(jj).
[> Line 180: ((intro);(subst)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const]
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R1(jj)) ||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R1(jj) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj) => R(jj', ii) <= R(jj0, ii)



use IH with pred(R1(jj)) => //.
[> Line 181: ((have);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const]
H: (forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > pred (R1(jj)))
||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= pred (R1(jj)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R1(jj)) =>
R(jj', ii) <= R(jj0, ii)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R1(jj)) ||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R1(jj) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj) => R(jj', ii) <= R(jj0, ii)



destruct H as [H1 | [j H2]].
[> Line 182: (destruct) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,jj:index[const]
H1: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > pred (R1(jj))
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R1(jj)) ||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R1(jj) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj) => R(jj', ii) <= R(jj0, ii)



- left; intro j HapT; by use H1 with j => //.
[> Line 183: (((left);((intro);by (have)));(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,j,jj:index[const]
H2: happens(R(j, ii)) &&
R(j, ii) <= pred (R1(jj)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R1(jj)) =>
R(jj', ii) <= R(j, ii)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R1(jj)) ||
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R1(jj) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj) => R(jj', ii) <= R(jj0, ii)



- right.
[> Line 183: (right) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,j,jj:index[const]
H2: happens(R(j, ii)) &&
R(j, ii) <= pred (R1(jj)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R1(jj)) =>
R(jj', ii) <= R(j, ii)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
----------------------------------------
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R1(jj) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj) => R(jj', ii) <= R(jj0, ii)


destruct H2 as [_ _ H2]. [> Line 184: (destruct) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,j,jj:index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R1(jj)) =>
R(jj', ii) <= R(j, ii)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
_: R(j, ii) <= pred (R1(jj))
_: happens(R(j, ii))
----------------------------------------
exists (jj0:index),
happens(R(jj0, ii)) &&
R(jj0, ii) <= R1(jj) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj) => R(jj', ii) <= R(jj0, ii)



exists j.
[> Line 185: (exists) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,j,jj:index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R1(jj)) =>
R(jj', ii) <= R(j, ii)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
_: R(j, ii) <= pred (R1(jj))
_: happens(R(j, ii))
----------------------------------------
happens(R(j, ii)) &&
R(j, ii) <= R1(jj) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj) => R(jj', ii) <= R(j, ii)



intro /= _ _.
[> Line 186: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,j,jj,jj':index[const]
H2: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R1(jj)) =>
R(jj', ii) <= R(j, ii)
Hap: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
_: happens(R(jj', ii)) && R(jj', ii) <= R1(jj)
_: R(j, ii) <= pred (R1(jj))
_: happens(R(j, ii))
----------------------------------------
R(jj', ii) <= R(j, ii)


by apply H2.
[> Line 187: by (apply) [goal> lemma lastupdate_pure_reader is proved


Qed.
lemma lastupdate_pure_reader @system:(set:default; equiv:None) :
forall (ii:index,tau:timestamp),
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) ||
exists (jj:index),
happens(R(jj, ii)) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Exiting proof mode.



lemma lastupdate_init_tag (i:index,tau:timestamp):
happens(tau) => (
(forall j, happens(T(i,j)) => T(i,j)>tau))
=> sT(i)@tau = sT(i)@init.
Goal lastupdate_init_tag :
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i:index,tau:timestamp
----------------------------------------
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) => sT i@tau = sT i@init


induction tau => tau IH Htau.
[> Line 195: ((induction);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) =>
sT i@tau0 = sT i@init
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) => sT i@tau = sT i@init


case tau.
[> Line 196: (case) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) =>
sT i@tau0 = sT i@init
----------------------------------------
tau = init =>
(forall (j:index), happens(T(i, j)) => T(i, j) > init) =>
sT i@init = sT i@init



(* init *)
+ auto.
[> Line 199: (auto) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) =>
sT i@tau0 = sT i@init
----------------------------------------
(exists (i0,j:index), tau = T(i0, j)) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) => sT i@tau = sT i@init



(* T(i0,j) *)
+ intro [i0 j HT]; rewrite HT in *.
[> Line 202: ((intro);(rewrite)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const],tau:timestamp[const]
HT: tau = T(i0, j)
Htau: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init
----------------------------------------
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) =>
sT i@T(i0, j) = sT i@init


case (i = i0).
[> Line 203: (case) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const],tau:timestamp[const]
HT: tau = T(i0, j)
Htau: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init
----------------------------------------
i = i0 =>
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) =>
sT i@T(i0, j) = sT i@init


- intro Eq H0.
[> Line 204: (intro) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const],tau:timestamp[const]
Eq: i = i0
H0: forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)
HT: tau = T(i0, j)
Htau: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init
----------------------------------------
sT i@T(i0, j) = sT i@init


by use H0 with j.
[> Line 205: by (have) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const],tau:timestamp[const]
HT: tau = T(i0, j)
Htau: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init
----------------------------------------
not (i = i0) =>
(forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)) =>
sT i@T(i0, j) = sT i@init


- intro Neq H0.
[> Line 206: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const],tau:timestamp[const]
H0: forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)
HT: tau = T(i0, j)
Htau: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init
Neq: not (i = i0)
----------------------------------------
sT i@T(i0, j) = sT i@init


use IH with pred(T(i0,j)) => //.
[> Line 207: ((have);(intro)) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const],tau:timestamp[const]
H0: forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)
HT: tau = T(i0, j)
Htau: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init
Meq: sT i@pred (T(i0, j)) = sT i@init
Neq: not (i = i0)
----------------------------------------
sT i@T(i0, j) = sT i@init


expand sT.
[> Line 208: (expand) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const],tau:timestamp[const]
H0: forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)
HT: tau = T(i0, j)
Htau: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = s0 i
Meq: sT i@pred (T(i0, j)) = s0 i
Neq: not (i = i0)
----------------------------------------
if (i = i0) then H (sT i0@pred (T(i0, j)), k) else sT i@pred (T(i0, j))
= s0 i


rewrite if_false //.
[> Line 208: (rewrite) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j:index[const],tau:timestamp[const]
H0: forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)
HT: tau = T(i0, j)
Htau: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init
Neq: not (i = i0)
----------------------------------------
forall (j0:index), happens(T(i, j0)) => T(i, j0) > pred (T(i0, j))


intro j0 Hp.
[> Line 210: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
H0: forall (j0:index), happens(T(i, j0)) => T(i, j0) > T(i0, j)
HT: tau = T(i0, j)
Hp: happens(T(i, j0))
Htau: happens(T(i0, j))
IH: forall (tau:timestamp),
tau < T(i0, j) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init
Neq: not (i = i0)
----------------------------------------
T(i, j0) > pred (T(i0, j))


by use H0 with j0.
[> Line 211: by (have) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) =>
sT i@tau0 = sT i@init
----------------------------------------
(exists (jj,ii:index), tau = R(jj, ii)) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) => sT i@tau = sT i@init



(* R(jj,ii) *)
+ intro HR.
[> Line 214: (intro) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
HR: exists (jj,ii:index), tau = R(jj, ii)
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) =>
sT i@tau0 = sT i@init
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) => sT i@tau = sT i@init


repeat destruct HR as [_ HR].
[> Line 215: (repeat (destruct)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii)
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) =>
sT i@tau0 = sT i@init
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) => sT i@tau = sT i@init


rewrite HR /sT in *.
[> Line 216: (rewrite) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii)
Htau: happens(R(jj, ii))
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = s0 i
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R(jj, ii)) =>
sT i@pred (R(jj, ii)) = s0 i


intro Hyp.
[> Line 217: (intro) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii)
Htau: happens(R(jj, ii))
Hyp: forall (j:index), happens(T(i, j)) => T(i, j) > R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = s0 i
----------------------------------------
sT i@pred (R(jj, ii)) = s0 i


apply IH => //.
[> Line 218: ((apply);(intro)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii)
Htau: happens(R(jj, ii))
Hyp: forall (j:index), happens(T(i, j)) => T(i, j) > R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = s0 i
----------------------------------------
forall (j:index), happens(T(i, j)) => T(i, j) > pred (R(jj, ii))


intro j _.
[> Line 219: (intro) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii)
Htau: happens(R(jj, ii))
Hyp: forall (j:index), happens(T(i, j)) => T(i, j) > R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = s0 i
_: happens(T(i, j))
----------------------------------------
T(i, j) > pred (R(jj, ii))


by use Hyp with j.
[> Line 220: by (have) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) =>
sT i@tau0 = sT i@init
----------------------------------------
(exists (jj:index), tau = R1(jj)) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) => sT i@tau = sT i@init



(* R1(jj) *)
(* same proof script as above *)
+ intro HR.
[> Line 224: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
HR: exists (jj:index), tau = R1(jj)
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) =>
sT i@tau0 = sT i@init
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) => sT i@tau = sT i@init


repeat destruct HR as [_ HR].
[> Line 225: (repeat (destruct)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,jj:index[const],tau:timestamp[const]
HR: tau = R1(jj)
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau0) =>
sT i@tau0 = sT i@init
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) => sT i@tau = sT i@init


rewrite HR /sT in *.
[> Line 226: (rewrite) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,jj:index[const],tau:timestamp[const]
HR: tau = R1(jj)
Htau: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = s0 i
----------------------------------------
(forall (j:index), happens(T(i, j)) => T(i, j) > R1(jj)) =>
sT i@pred (R1(jj)) = s0 i


intro Hyp.
[> Line 227: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,jj:index[const],tau:timestamp[const]
HR: tau = R1(jj)
Htau: happens(R1(jj))
Hyp: forall (j:index), happens(T(i, j)) => T(i, j) > R1(jj)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = s0 i
----------------------------------------
sT i@pred (R1(jj)) = s0 i


apply IH => //.
[> Line 228: ((apply);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,jj:index[const],tau:timestamp[const]
HR: tau = R1(jj)
Htau: happens(R1(jj))
Hyp: forall (j:index), happens(T(i, j)) => T(i, j) > R1(jj)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = s0 i
----------------------------------------
forall (j:index), happens(T(i, j)) => T(i, j) > pred (R1(jj))


intro j _.
[> Line 229: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j,jj:index[const],tau:timestamp[const]
HR: tau = R1(jj)
Htau: happens(R1(jj))
Hyp: forall (j:index), happens(T(i, j)) => T(i, j) > R1(jj)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = s0 i
_: happens(T(i, j))
----------------------------------------
T(i, j) > pred (R1(jj))


by use Hyp with j.
[> Line 230: by (have) [goal> lemma lastupdate_init_tag is proved


Qed.
lemma lastupdate_init_tag @system:(set:default; equiv:None) :
forall (i:index,tau:timestamp),
happens(tau) =>
(forall (j:index), happens(T(i, j)) => T(i, j) > tau) =>
sT i@tau = sT i@init
Exiting proof mode.



lemma lastupdate_init_reader (ii:index,tau:timestamp):
happens(tau) => (
(forall jj, happens(R(jj,ii)) => R(jj,ii)>tau))
=> sR(ii)@tau = sR(ii)@init.
Goal lastupdate_init_reader :
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii:index,tau:timestamp
----------------------------------------
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init


induction tau => tau IH Htau.
[> Line 238: ((induction);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) =>
sR ii@tau0 = sR ii@init
----------------------------------------
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init


case tau.
[> Line 239: (case) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) =>
sR ii@tau0 = sR ii@init
----------------------------------------
tau = init =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > init) =>
sR ii@init = sR ii@init



(* init *)
+ auto.
[> Line 242: (auto) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) =>
sR ii@tau0 = sR ii@init
----------------------------------------
(exists (i,j:index), tau = T(i, j)) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init



(* T(i,j) *)
+ intro [i j HT]; rewrite HT in *.
[> Line 245: ((intro);(rewrite)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j:index[const],tau:timestamp[const]
HT: tau = T(i, j)
Htau: happens(T(i, j))
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > T(i, j)) =>
sR ii@T(i, j) = sR ii@init


intro @/sR Hyp.
[> Line 246: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j:index[const],tau:timestamp[const]
HT: tau = T(i, j)
Htau: happens(T(i, j))
Hyp: forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > T(i, j)
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
sR ii@pred (T(i, j)) = s0 ii


use IH with pred(T(i,j)) => //.
[> Line 247: ((have);(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j:index[const],tau:timestamp[const]
HT: tau = T(i, j)
Htau: happens(T(i, j))
Hyp: forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > T(i, j)
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > pred (T(i, j))


intro jj _.
[> Line 248: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const],tau:timestamp[const]
HT: tau = T(i, j)
Htau: happens(T(i, j))
Hyp: forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > T(i, j)
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
_: happens(R(jj, ii))
----------------------------------------
R(jj, ii) > pred (T(i, j))


by use Hyp with jj.
[> Line 249: by (have) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) =>
sR ii@tau0 = sR ii@init
----------------------------------------
(exists (jj,ii0:index), tau = R(jj, ii0)) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init



(* R(jj,ii0) *)
+ intro [jj ii0 HR]; rewrite HR in *.
[> Line 252: ((intro);(rewrite)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii0)
Htau: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
=> sR ii@R(jj, ii0) = sR ii@init


case (ii = ii0) => //.
[> Line 253: ((case);(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii0)
Htau: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
ii = ii0 =>
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
=> sR ii@R(jj, ii0) = sR ii@init


- intro Eq H0.
[> Line 254: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const],tau:timestamp[const]
Eq: ii = ii0
H0: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0)
HR: tau = R(jj, ii0)
Htau: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
sR ii@R(jj, ii0) = sR ii@init


by use H0 with jj.
[> Line 255: by (have) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii0)
Htau: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
not (ii = ii0) =>
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0))
=> sR ii@R(jj, ii0) = sR ii@init


- intro Neq H0.
[> Line 256: (intro) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const],tau:timestamp[const]
H0: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0)
HR: tau = R(jj, ii0)
Htau: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
Neq: not (ii = ii0)
----------------------------------------
sR ii@R(jj, ii0) = sR ii@init


use IH with pred(R(jj,ii0)) => //.
[> Line 257: ((have);(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const],tau:timestamp[const]
H0: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0)
HR: tau = R(jj, ii0)
Htau: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
Meq: sR ii@pred (R(jj, ii0)) = sR ii@init
Neq: not (ii = ii0)
----------------------------------------
sR ii@R(jj, ii0) = sR ii@init


expand sR.
[> Line 258: (expand) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const],tau:timestamp[const]
H0: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0)
HR: tau = R(jj, ii0)
Htau: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = s0 ii
Meq: sR ii@pred (R(jj, ii0)) = s0 ii
Neq: not (ii = ii0)
----------------------------------------
if (ii = ii0) then H (sR ii0@pred (R(jj, ii0)), k)
else sR ii@pred (R(jj, ii0)) = s0 ii


rewrite if_false //.
[> Line 259: (rewrite) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj:index[const],tau:timestamp[const]
H0: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0)
HR: tau = R(jj, ii0)
Htau: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
Neq: not (ii = ii0)
----------------------------------------
forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > pred (R(jj, ii0))


intro jj0 Hp.
[> Line 260: (intro) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
H0: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R(jj, ii0)
HR: tau = R(jj, ii0)
Hp: happens(R(jj0, ii))
Htau: happens(R(jj, ii0))
IH: forall (tau:timestamp),
tau < R(jj, ii0) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
Neq: not (ii = ii0)
----------------------------------------
R(jj0, ii) > pred (R(jj, ii0))


by use H0 with jj0.
[> Line 261: by (have) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Htau: happens(tau)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau0) =>
sR ii@tau0 = sR ii@init
----------------------------------------
(exists (jj:index), tau = R1(jj)) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init



(* R1(jj) *)
+ intro [jj HR1]; rewrite HR1 in *.
[> Line 264: ((intro);(rewrite)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
HR1: tau = R1(jj)
Htau: happens(R1(jj))
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
(forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R1(jj)) =>
sR ii@R1(jj) = sR ii@init


intro @/sR Hyp.
[> Line 265: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
HR1: tau = R1(jj)
Htau: happens(R1(jj))
Hyp: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R1(jj)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
sR ii@pred (R1(jj)) = s0 ii


use IH with pred(R1(jj)) => //.
[> Line 266: ((have);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
HR1: tau = R1(jj)
Htau: happens(R1(jj))
Hyp: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R1(jj)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
----------------------------------------
forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > pred (R1(jj))


intro jj0 _.
[> Line 267: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj,jj0:index[const],tau:timestamp[const]
HR1: tau = R1(jj)
Htau: happens(R1(jj))
Hyp: forall (jj0:index), happens(R(jj0, ii)) => R(jj0, ii) > R1(jj)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
_: happens(R(jj0, ii))
----------------------------------------
R(jj0, ii) > pred (R1(jj))


by use Hyp with jj0.
[> Line 268: by (have) [goal> lemma lastupdate_init_reader is proved


Qed.
lemma lastupdate_init_reader @system:(set:default; equiv:None) :
forall (ii:index,tau:timestamp),
happens(tau) =>
(forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau) =>
sR ii@tau = sR ii@init
Exiting proof mode.



lemma lastupdate_T (i:index, j:index, tau:timestamp):
(happens(tau) && T(i,j)<=tau &&
forall j', happens(T(i,j')) && T(i,j')<=tau => T(i,j')<=T(i,j))
=> sT(i)@tau = sT(i)@T(i,j).
Goal lastupdate_T :
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j:index,tau:timestamp
----------------------------------------
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)


induction tau => tau IH [Hp Ord Hyp].
[> Line 276: ((induction);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
=> sT i@tau0 = sT i@T(i, j)
Ord: T(i, j) <= tau
----------------------------------------
sT i@tau = sT i@T(i, j)


case tau.
[> Line 277: (case) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,j:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
=> sT i@tau0 = sT i@T(i, j)
Ord: T(i, j) <= tau
----------------------------------------
tau = init => sT i@init = sT i@T(i, j)



(* init *)
- auto.
[> Line 280: (auto) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,j:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
=> sT i@tau0 = sT i@T(i, j)
Ord: T(i, j) <= tau
----------------------------------------
(exists (i0,j0:index), tau = T(i0, j0)) => sT i@tau = sT i@T(i, j)



(* T(i0,j0) *)
- intro [i0 j0 H]; rewrite H in *.
[> Line 283: ((intro);(rewrite)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Ord: T(i, j) <= T(i0, j0)
----------------------------------------
sT i@T(i0, j0) = sT i@T(i, j)


case (i=i0).
[> Line 284: (case) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Ord: T(i, j) <= T(i0, j0)
----------------------------------------
i = i0 => sT i@T(i0, j0) = sT i@T(i, j)


+ intro Eqi.
[> Line 285: (intro) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
Eqi: i = i0
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Ord: T(i, j) <= T(i0, j0)
----------------------------------------
sT i@T(i0, j0) = sT i@T(i, j)


use Hyp with j0 => //.
[> Line 286: ((have);(intro)) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
Cle: T(i, j0) <= T(i, j)
Eqi: i = i0
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Ord: T(i, j) <= T(i0, j0)
----------------------------------------
sT i@T(i0, j0) = sT i@T(i, j)


case (j=j0); [1: by rewrite if_true | 2:auto].
[> Line 287: ((case); [1: by (rewrite)|2: (auto)]) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Ord: T(i, j) <= T(i0, j0)
----------------------------------------
not (i = i0) => sT i@T(i0, j0) = sT i@T(i, j)



+ intro Neqi.
[> Line 289: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Neqi: not (i = i0)
Ord: T(i, j) <= T(i0, j0)
----------------------------------------
sT i@T(i0, j0) = sT i@T(i, j)


expand sT(i)@T(i0,j0).
[> Line 290: (expand) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Neqi: not (i = i0)
Ord: T(i, j) <= T(i0, j0)
----------------------------------------
if (i = i0) then H (sT i0@pred (T(i0, j0)), k) else sT i@pred (T(i0, j0))
= sT i@T(i, j)


rewrite if_false //.
[> Line 291: (rewrite) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Neqi: not (i = i0)
Ord: T(i, j) <= T(i0, j0)
----------------------------------------
sT i@pred (T(i0, j0)) = sT i@T(i, j)


use IH with pred(T(i0,j0)) => //.
[> Line 292: ((have);(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j,j0:index[const],tau:timestamp[const]
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Neqi: not (i = i0)
Ord: T(i, j) <= T(i0, j0)
----------------------------------------
happens(pred (T(i0, j0))) &&
T(i, j) <= pred (T(i0, j0)) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (T(i0, j0)) => T(i, j') <= T(i, j)



intro /= _ _.
[> Line 293: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i0,j,j',j0:index[const],tau:timestamp[const]
H: tau = T(i0, j0)
Hp: happens(T(i0, j0))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= T(i0, j0) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < T(i0, j0) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Neqi: i <> i0
Ord: T(i, j) <= T(i0, j0)
_: happens(T(i, j')) && T(i, j') <= pred (T(i0, j0))
----------------------------------------
T(i, j') <= T(i, j)


by apply Hyp.
[> Line 294: by (apply) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,j:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
=> sT i@tau0 = sT i@T(i, j)
Ord: T(i, j) <= tau
----------------------------------------
(exists (jj,ii:index), tau = R(jj, ii)) => sT i@tau = sT i@T(i, j)



(* R(jj,ii) *)
- intro [jj ii HR]; rewrite HR in *.
[> Line 297: ((intro);(rewrite)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii)
Hp: happens(R(jj, ii))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Ord: T(i, j) <= R(jj, ii)
----------------------------------------
sT i@R(jj, ii) = sT i@T(i, j)


expand sT.
[> Line 298: (expand) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii)
Hp: happens(R(jj, ii))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = H (sT i@pred (T(i, j)), k)
Ord: T(i, j) <= R(jj, ii)
----------------------------------------
sT i@pred (R(jj, ii)) = H (sT i@pred (T(i, j)), k)


use IH with pred(R(jj,ii)) => //.
[> Line 299: ((have);(intro)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii)
Hp: happens(R(jj, ii))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = H (sT i@pred (T(i, j)), k)
Ord: T(i, j) <= R(jj, ii)
----------------------------------------
happens(pred (R(jj, ii))) &&
T(i, j) <= pred (R(jj, ii)) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R(jj, ii)) => T(i, j') <= T(i, j)



intro /= _ _.
[> Line 300: (intro) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,ii,j,j',jj:index[const],tau:timestamp[const]
HR: tau = R(jj, ii)
Hp: happens(R(jj, ii))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= R(jj, ii) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < R(jj, ii) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = H (sT i@pred (T(i, j)), k)
Ord: T(i, j) <= R(jj, ii)
_: happens(T(i, j')) && T(i, j') <= pred (R(jj, ii))
----------------------------------------
T(i, j') <= T(i, j)


by apply Hyp.
[> Line 301: by (apply) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
T(i, j) <= tau0 &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau0 => T(i, j') <= T(i, j)
=> sT i@tau0 = sT i@T(i, j)
Ord: T(i, j) <= tau
----------------------------------------
(exists (jj:index), tau = R1(jj)) => sT i@tau = sT i@T(i, j)



(* R1(jj) *)
- intro [jj HR1]; rewrite HR1 in *.
[> Line 304: ((intro);(rewrite)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j,jj:index[const],tau:timestamp[const]
HR1: tau = R1(jj)
Hp: happens(R1(jj))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Ord: T(i, j) <= R1(jj)
----------------------------------------
sT i@R1(jj) = sT i@T(i, j)


expand sT.
[> Line 305: (expand) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j,jj:index[const],tau:timestamp[const]
HR1: tau = R1(jj)
Hp: happens(R1(jj))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = H (sT i@pred (T(i, j)), k)
Ord: T(i, j) <= R1(jj)
----------------------------------------
sT i@pred (R1(jj)) = H (sT i@pred (T(i, j)), k)


use IH with pred(R1(jj)) => //.
[> Line 306: ((have);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j,jj:index[const],tau:timestamp[const]
HR1: tau = R1(jj)
Hp: happens(R1(jj))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = H (sT i@pred (T(i, j)), k)
Ord: T(i, j) <= R1(jj)
----------------------------------------
happens(pred (R1(jj))) &&
T(i, j) <= pred (R1(jj)) &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= pred (R1(jj)) => T(i, j') <= T(i, j)



intro /= _ _.
[> Line 307: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j,j',jj:index[const],tau:timestamp[const]
HR1: tau = R1(jj)
Hp: happens(R1(jj))
Hyp: forall (j':index),
happens(T(i, j')) && T(i, j') <= R1(jj) => T(i, j') <= T(i, j)
IH: forall (tau:timestamp),
tau < R1(jj) =>
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = H (sT i@pred (T(i, j)), k)
Ord: T(i, j) <= R1(jj)
_: happens(T(i, j')) && T(i, j') <= pred (R1(jj))
----------------------------------------
T(i, j') <= T(i, j)


by apply Hyp.
[> Line 308: by (apply) [goal> lemma lastupdate_T is proved


Qed.
lemma lastupdate_T @system:(set:default; equiv:None) :
forall (i,j:index,tau:timestamp),
happens(tau) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j) =>
sT i@tau = sT i@T(i, j)
Exiting proof mode.



lemma lastupdate_R (ii:index, jj:index, tau:timestamp):
(happens(tau) && R(jj,ii)<=tau &&
forall jj',
happens(R(jj',ii)) && R(jj',ii)<=tau => R(jj',ii)<=R(jj,ii))
=> sR(ii)@tau = sR(ii)@R(jj,ii).
Goal lastupdate_R :
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index,tau:timestamp
----------------------------------------
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)


induction tau => tau IH [Hp Ord Hyp].
[> Line 317: ((induction);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 => R(jj', ii) <= R(jj, ii)
=> sR ii@tau0 = sR ii@R(jj, ii)
Ord: R(jj, ii) <= tau
----------------------------------------
sR ii@tau = sR ii@R(jj, ii)


case tau.
[> Line 318: (case) [goal> Focused goal (1/4):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 => R(jj', ii) <= R(jj, ii)
=> sR ii@tau0 = sR ii@R(jj, ii)
Ord: R(jj, ii) <= tau
----------------------------------------
tau = init => sR ii@init = sR ii@R(jj, ii)



(* init *)
- auto.
[> Line 321: (auto) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 => R(jj', ii) <= R(jj, ii)
=> sR ii@tau0 = sR ii@R(jj, ii)
Ord: R(jj, ii) <= tau
----------------------------------------
(exists (i,j:index), tau = T(i, j)) => sR ii@tau = sR ii@R(jj, ii)



(* T(i,j) *)
- intro [i j HT]; rewrite HT in *.
[> Line 324: ((intro);(rewrite)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const],tau:timestamp[const]
HT: tau = T(i, j)
Hp: happens(T(i, j))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)
Ord: R(jj, ii) <= T(i, j)
----------------------------------------
sR ii@T(i, j) = sR ii@R(jj, ii)


expand sR.
[> Line 325: (expand) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const],tau:timestamp[const]
HT: tau = T(i, j)
Hp: happens(T(i, j))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Ord: R(jj, ii) <= T(i, j)
----------------------------------------
sR ii@pred (T(i, j)) = H (sR ii@pred (R(jj, ii)), k)


use IH with pred(T(i,j)) => //.
[> Line 326: ((have);(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const],tau:timestamp[const]
HT: tau = T(i, j)
Hp: happens(T(i, j))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Ord: R(jj, ii) <= T(i, j)
----------------------------------------
happens(pred (T(i, j))) &&
R(jj, ii) <= pred (T(i, j)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (T(i, j)) =>
R(jj', ii) <= R(jj, ii)



intro /= _ _.
[> Line 327: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,ii,j,jj,jj':index[const],tau:timestamp[const]
HT: tau = T(i, j)
Hp: happens(T(i, j))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= T(i, j) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < T(i, j) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Ord: R(jj, ii) <= T(i, j)
_: happens(R(jj', ii)) && R(jj', ii) <= pred (T(i, j))
----------------------------------------
R(jj', ii) <= R(jj, ii)


by apply Hyp.
[> Line 328: by (apply) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 => R(jj', ii) <= R(jj, ii)
=> sR ii@tau0 = sR ii@R(jj, ii)
Ord: R(jj, ii) <= tau
----------------------------------------
(exists (jj0,ii0:index), tau = R(jj0, ii0)) => sR ii@tau = sR ii@R(jj, ii)



(* R(jj0,ii0) *)
- intro [jj0 ii0 H]; rewrite H in *.
[> Line 331: ((intro);(rewrite)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)
Ord: R(jj, ii) <= R(jj0, ii0)
----------------------------------------
sR ii@R(jj0, ii0) = sR ii@R(jj, ii)


case (ii=ii0) => //.
[> Line 332: ((case);(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)
Ord: R(jj, ii) <= R(jj0, ii0)
----------------------------------------
ii = ii0 => sR ii@R(jj0, ii0) = sR ii@R(jj, ii)


+ intro Eqii.
[> Line 333: (intro) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
Eqii: ii = ii0
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)
Ord: R(jj, ii) <= R(jj0, ii0)
----------------------------------------
sR ii@R(jj0, ii0) = sR ii@R(jj, ii)


have ? := Hyp jj0 _; 1:auto.
[> Line 334: ((have); 1: (auto)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
Cle: R(jj0, ii) <= R(jj, ii)
Eqii: ii = ii0
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)
Ord: R(jj, ii) <= R(jj0, ii0)
----------------------------------------
sR ii@R(jj0, ii0) = sR ii@R(jj, ii)


case (jj=jj0); [1: by rewrite if_true | 2:auto].
[> Line 335: ((case); [1: by (rewrite)|2: (auto)]) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)
Ord: R(jj, ii) <= R(jj0, ii0)
----------------------------------------
not (ii = ii0) => sR ii@R(jj0, ii0) = sR ii@R(jj, ii)



+ intro Neqii.
[> Line 337: (intro) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)
Neqii: not (ii = ii0)
Ord: R(jj, ii) <= R(jj0, ii0)
----------------------------------------
sR ii@R(jj0, ii0) = sR ii@R(jj, ii)


expand sR.
[> Line 338: (expand) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Neqii: not (ii = ii0)
Ord: R(jj, ii) <= R(jj0, ii0)
----------------------------------------
if (ii = ii0) then H (sR ii0@pred (R(jj0, ii0)), k)
else sR ii@pred (R(jj0, ii0)) = H (sR ii@pred (R(jj, ii)), k)


rewrite if_false //.
[> Line 339: (rewrite) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Neqii: not (ii = ii0)
Ord: R(jj, ii) <= R(jj0, ii0)
----------------------------------------
sR ii@pred (R(jj0, ii0)) = H (sR ii@pred (R(jj, ii)), k)


use IH with pred(R(jj0,ii0)) => //.
[> Line 340: ((have);(intro)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj0:index[const],tau:timestamp[const]
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Neqii: not (ii = ii0)
Ord: R(jj, ii) <= R(jj0, ii0)
----------------------------------------
happens(pred (R(jj0, ii0))) &&
R(jj, ii) <= pred (R(jj0, ii0)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R(jj0, ii0)) =>
R(jj', ii) <= R(jj, ii)



intro /= _ _.
[> Line 341: (intro) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii,ii0,jj,jj',jj0:index[const],tau:timestamp[const]
H: tau = R(jj0, ii0)
Hp: happens(R(jj0, ii0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R(jj0, ii0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R(jj0, ii0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Neqii: ii <> ii0
Ord: R(jj, ii) <= R(jj0, ii0)
_: happens(R(jj', ii)) && R(jj', ii) <= pred (R(jj0, ii0))
----------------------------------------
R(jj', ii) <= R(jj, ii)


by apply Hyp.
[> Line 342: by (apply) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
Hp: happens(tau)
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
IH: forall (tau0:timestamp),
tau0 < tau =>
happens(tau0) &&
R(jj, ii) <= tau0 &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau0 => R(jj', ii) <= R(jj, ii)
=> sR ii@tau0 = sR ii@R(jj, ii)
Ord: R(jj, ii) <= tau
----------------------------------------
(exists (jj0:index), tau = R1(jj0)) => sR ii@tau = sR ii@R(jj, ii)



(* R1(jj0) *)
- intro [jj0 HR1]; rewrite HR1 in *.
[> Line 345: ((intro);(rewrite)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj,jj0:index[const],tau:timestamp[const]
HR1: tau = R1(jj0)
Hp: happens(R1(jj0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R1(jj0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)
Ord: R(jj, ii) <= R1(jj0)
----------------------------------------
sR ii@R1(jj0) = sR ii@R(jj, ii)


expand sR.
[> Line 346: (expand) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj,jj0:index[const],tau:timestamp[const]
HR1: tau = R1(jj0)
Hp: happens(R1(jj0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R1(jj0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Ord: R(jj, ii) <= R1(jj0)
----------------------------------------
sR ii@pred (R1(jj0)) = H (sR ii@pred (R(jj, ii)), k)


use IH with pred(R1(jj0)) => //.
[> Line 347: ((have);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj,jj0:index[const],tau:timestamp[const]
HR1: tau = R1(jj0)
Hp: happens(R1(jj0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R1(jj0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Ord: R(jj, ii) <= R1(jj0)
----------------------------------------
happens(pred (R1(jj0))) &&
R(jj, ii) <= pred (R1(jj0)) &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= pred (R1(jj0)) =>
R(jj', ii) <= R(jj, ii)



intro /= _ _.
[> Line 348: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj,jj',jj0:index[const],tau:timestamp[const]
HR1: tau = R1(jj0)
Hp: happens(R1(jj0))
Hyp: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= R1(jj0) =>
R(jj', ii) <= R(jj, ii)
IH: forall (tau:timestamp),
tau < R1(jj0) =>
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = H (sR ii@pred (R(jj, ii)), k)
Ord: R(jj, ii) <= R1(jj0)
_: happens(R(jj', ii)) && R(jj', ii) <= pred (R1(jj0))
----------------------------------------
R(jj', ii) <= R(jj, ii)


by apply Hyp.
[> Line 349: by (apply) [goal> lemma lastupdate_R is proved


Qed.
lemma lastupdate_R @system:(set:default; equiv:None) :
forall (ii,jj:index,tau:timestamp),
happens(tau) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
=> sR ii@tau = sR ii@R(jj, ii)
Exiting proof mode.



lemma lastupdateTag (i:index,tau:timestamp):
happens(tau) => (
(sT(i)@tau = sT(i)@init && forall j, happens(T(i,j)) => T(i,j)>tau) ||
(exists j,
sT(i)@tau = sT(i)@T(i,j) && T(i,j)<=tau &&
forall j', happens(T(i,j')) && T(i,j')<=tau => T(i,j')<=T(i,j))).
Goal lastupdateTag :
happens(tau) =>
sT i@tau = sT i@init && forall (j:index), happens(T(i, j)) => T(i, j) > tau
||
exists (j:index),
sT i@tau = sT i@T(i, j) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i:index,tau:timestamp
----------------------------------------
happens(tau) =>
sT i@tau = sT i@init && forall (j:index), happens(T(i, j)) => T(i, j) > tau
||
exists (j:index),
sT i@tau = sT i@T(i, j) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)



intro Htau.
[> Line 359: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Htau: happens(tau)
----------------------------------------
sT i@tau = sT i@init && forall (j:index), happens(T(i, j)) => T(i, j) > tau
||
exists (j:index),
sT i@tau = sT i@T(i, j) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)



use lastupdate_pure_tag with i, tau as [Hinit | [j [HTj1 HTj2 HTj3]]] => //.
[> Line 360: ((have);(intro)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Hinit: forall (j:index), happens(T(i, j)) => T(i, j) > tau
Htau: happens(tau)
----------------------------------------
sT i@tau = sT i@init && forall (j:index), happens(T(i, j)) => T(i, j) > tau
||
exists (j:index),
sT i@tau = sT i@T(i, j) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)



- left.
[> Line 361: (left) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Hinit: forall (j:index), happens(T(i, j)) => T(i, j) > tau
Htau: happens(tau)
----------------------------------------
sT i@tau = sT i@init && forall (j:index), happens(T(i, j)) => T(i, j) > tau


simpl.
[> Line 362: (simpl) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i:index[const],tau:timestamp[const]
Hinit: forall (j:index), happens(T(i, j)) => T(i, j) > tau
Htau: happens(tau)
----------------------------------------
sT i@tau = sT i@init


by apply lastupdate_init_tag.
[> Line 363: by (apply) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j:index[const],tau:timestamp[const]
HTj1: happens(T(i, j))
HTj2: T(i, j) <= tau
HTj3: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Htau: happens(tau)
----------------------------------------
sT i@tau = sT i@init && forall (j:index), happens(T(i, j)) => T(i, j) > tau
||
exists (j:index),
sT i@tau = sT i@T(i, j) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)



- right.
[> Line 364: (right) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j:index[const],tau:timestamp[const]
HTj1: happens(T(i, j))
HTj2: T(i, j) <= tau
HTj3: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Htau: happens(tau)
----------------------------------------
exists (j:index),
sT i@tau = sT i@T(i, j) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)



exists j.
[> Line 365: (exists) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j:index[const],tau:timestamp[const]
HTj1: happens(T(i, j))
HTj2: T(i, j) <= tau
HTj3: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Htau: happens(tau)
----------------------------------------
sT i@tau = sT i@T(i, j) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)



simpl.
[> Line 366: (simpl) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,j:index[const],tau:timestamp[const]
HTj1: happens(T(i, j))
HTj2: T(i, j) <= tau
HTj3: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Htau: happens(tau)
----------------------------------------
sT i@tau = sT i@T(i, j)


by apply lastupdate_T.
[> Line 367: by (apply) [goal> lemma lastupdateTag is proved


Qed.
lemma lastupdateTag @system:(set:default; equiv:None) :
forall (i:index,tau:timestamp),
happens(tau) =>
sT i@tau = sT i@init &&
forall (j:index), happens(T(i, j)) => T(i, j) > tau ||
exists (j:index),
sT i@tau = sT i@T(i, j) &&
T(i, j) <= tau &&
forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Exiting proof mode.



lemma lastupdateReader (ii:index,tau:timestamp):
happens(tau) => (
(sR(ii)@tau = sR(ii)@init &&
forall jj, happens(R(jj,ii)) => R(jj,ii)>tau) ||
(exists jj,
sR(ii)@tau = sR(ii)@R(jj,ii) && R(jj,ii)<=tau &&
forall jj',
happens(R(jj',ii)) && R(jj',ii)<=tau => R(jj',ii)<=R(jj,ii))).
Goal lastupdateReader :
happens(tau) =>
sR ii@tau = sR ii@init &&
forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau ||
exists (jj:index),
sR ii@tau = sR ii@R(jj, ii) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii:index,tau:timestamp
----------------------------------------
happens(tau) =>
sR ii@tau = sR ii@init &&
forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau ||
exists (jj:index),
sR ii@tau = sR ii@R(jj, ii) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)



intro Htau.
[> Line 379: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Htau: happens(tau)
----------------------------------------
sR ii@tau = sR ii@init &&
forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau ||
exists (jj:index),
sR ii@tau = sR ii@R(jj, ii) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)



use lastupdate_pure_reader with ii, tau as [Hinit | [jj [HTj1 HTj2 HTj3]]] => //.
[> Line 380: ((have);(intro)) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Hinit: forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau
Htau: happens(tau)
----------------------------------------
sR ii@tau = sR ii@init &&
forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau ||
exists (jj:index),
sR ii@tau = sR ii@R(jj, ii) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)



- left.
[> Line 381: (left) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Hinit: forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau
Htau: happens(tau)
----------------------------------------
sR ii@tau = sR ii@init &&
forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau


simpl.
[> Line 382: (simpl) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: ii:index[const],tau:timestamp[const]
Hinit: forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau
Htau: happens(tau)
----------------------------------------
sR ii@tau = sR ii@init


by apply lastupdate_init_reader.
[> Line 383: by (apply) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
HTj1: happens(R(jj, ii))
HTj2: R(jj, ii) <= tau
HTj3: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Htau: happens(tau)
----------------------------------------
sR ii@tau = sR ii@init &&
forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau ||
exists (jj:index),
sR ii@tau = sR ii@R(jj, ii) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)



- right.
[> Line 384: (right) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
HTj1: happens(R(jj, ii))
HTj2: R(jj, ii) <= tau
HTj3: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Htau: happens(tau)
----------------------------------------
exists (jj:index),
sR ii@tau = sR ii@R(jj, ii) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)



exists jj.
[> Line 385: (exists) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
HTj1: happens(R(jj, ii))
HTj2: R(jj, ii) <= tau
HTj3: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Htau: happens(tau)
----------------------------------------
sR ii@tau = sR ii@R(jj, ii) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)



simpl.
[> Line 386: (simpl) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const],tau:timestamp[const]
HTj1: happens(R(jj, ii))
HTj2: R(jj, ii) <= tau
HTj3: forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Htau: happens(tau)
----------------------------------------
sR ii@tau = sR ii@R(jj, ii)


by apply lastupdate_R.
[> Line 387: by (apply) [goal> lemma lastupdateReader is proved


Qed.
lemma lastupdateReader @system:(set:default; equiv:None) :
forall (ii:index,tau:timestamp),
happens(tau) =>
sR ii@tau = sR ii@init &&
forall (jj:index), happens(R(jj, ii)) => R(jj, ii) > tau ||
exists (jj:index),
sR ii@tau = sR ii@R(jj, ii) &&
R(jj, ii) <= tau &&
forall (jj':index),
happens(R(jj', ii)) && R(jj', ii) <= tau => R(jj', ii) <= R(jj, ii)
Exiting proof mode.



(* The following lemma states that values of different memory cells do not
overlap, relying on the collision resistance of the hash function. *)

lemma disjoint_chains (tau',tau:timestamp,i',i:index) :
happens(tau',tau) =>
i<>i' => sT(i)@tau <> sR(i')@tau'.
Goal disjoint_chains :
happens(tau, tau') => i <> i' => sT i@tau <> sR i'@tau'

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,i':index,tau,tau':timestamp
----------------------------------------
happens(tau, tau') => i <> i' => sT i@tau <> sR i'@tau'


generalize tau.
[> Line 397: (generalize) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,i':index[const],tau':timestamp[const]
----------------------------------------
forall (tau:timestamp),
happens(tau, tau') => i <> i' => sT i@tau <> sR i'@tau'



induction tau' => tau' IH tau D E Meq.
[> Line 398: ((induction);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,i':index[const],tau,tau':timestamp[const]
D: happens(tau, tau')
E: i <> i'
IH: forall (tau'0,tau:timestamp),
tau'0 < tau' =>
happens(tau, tau'0) => i <> i' => sT i@tau <> sR i'@tau'0
Meq: sT i@tau = sR i'@tau'
----------------------------------------
false


use lastupdateTag with i,tau as [[A0 Hinit] | [j [A0 A1 Hsup]]] => //;
use lastupdateReader with i',tau' as [[A Hinit'] | [j' [B C Hsup']]] => //.
[> Line 400: ((((have);(intro));(have));(intro)) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i',j':index[const],tau,tau':timestamp[const]
A0: sT i@tau = sT i@init
B: sR i'@tau' = sR i'@R(j', i')
C: R(j', i') <= tau'
D: happens(tau, tau')
E: i <> i'
Hinit: forall (j:index), happens(T(i, j)) => T(i, j) > tau
Hsup': forall (jj':index),
happens(R(jj', i')) && R(jj', i') <= tau' => R(jj', i') <= R(j', i')
IH: forall (tau'0,tau:timestamp),
tau'0 < tau' =>
happens(tau, tau'0) => i <> i' => sT i@tau <> sR i'@tau'0
Meq: sT i@tau = sR i'@tau'
----------------------------------------
false



+ rewrite -Meq A0 /sR in B.
[> Line 402: (rewrite) [goal> Focused goal (1/3):
System: (set:default; equiv:None)
Variables: i,i',j':index[const],tau,tau':timestamp[const]
A0: sT i@tau = sT i@init
B: sT i@init = H (sR i'@pred (R(j', i')), k)
C: R(j', i') <= tau'
D: happens(tau, tau')
E: i <> i'
Hinit: forall (j:index), happens(T(i, j)) => T(i, j) > tau
Hsup': forall (jj':index),
happens(R(jj', i')) && R(jj', i') <= tau' => R(jj', i') <= R(j', i')
IH: forall (tau'0,tau:timestamp),
tau'0 < tau' =>
happens(tau, tau'0) => i <> i' => sT i@tau <> sR i'@tau'0
Meq: sT i@tau = sR i'@tau'
----------------------------------------
false


by fresh B.
[> Line 403: by (fresh) Freshness of s0 i:
Indirect occurrences of s0(i) in other actions:
s0(i')
(collision with s0(i))
in action init
in term (happens(init), s0 i')

Total: 1 occurrence
0 of them are subsumed by another
1 occurrence remaining


[goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,i',j:index[const],tau,tau':timestamp[const]
A: sR i'@tau' = sR i'@init
A0: sT i@tau = sT i@T(i, j)
A1: T(i, j) <= tau
D: happens(tau, tau')
E: i <> i'
Hinit': forall (jj:index), happens(R(jj, i')) => R(jj, i') > tau'
Hsup: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
IH: forall (tau'0,tau:timestamp),
tau'0 < tau' =>
happens(tau, tau'0) => i <> i' => sT i@tau <> sR i'@tau'0
Meq: sT i@tau = sR i'@tau'
----------------------------------------
false



+ rewrite Meq A /sT in A0.
[> Line 405: (rewrite) [goal> Focused goal (1/2):
System: (set:default; equiv:None)
Variables: i,i',j:index[const],tau,tau':timestamp[const]
A: sR i'@tau' = sR i'@init
A0: sR i'@init = H (sT i@pred (T(i, j)), k)
A1: T(i, j) <= tau
D: happens(tau, tau')
E: i <> i'
Hinit': forall (jj:index), happens(R(jj, i')) => R(jj, i') > tau'
Hsup: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
IH: forall (tau'0,tau:timestamp),
tau'0 < tau' =>
happens(tau, tau'0) => i <> i' => sT i@tau <> sR i'@tau'0
Meq: sT i@tau = sR i'@tau'
----------------------------------------
false


by fresh A0.
[> Line 406: by (fresh) Freshness of s0 i':
Indirect occurrences of s0(i') in other actions:
s0(i) (collision with s0(i')) in action init in term (happens(init), s0 i)

Total: 1 occurrence
0 of them are subsumed by another
1 occurrence remaining


[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,i',j,j':index[const],tau,tau':timestamp[const]
A0: sT i@tau = sT i@T(i, j)
A1: T(i, j) <= tau
B: sR i'@tau' = sR i'@R(j', i')
C: R(j', i') <= tau'
D: happens(tau, tau')
E: i <> i'
Hsup: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Hsup': forall (jj':index),
happens(R(jj', i')) && R(jj', i') <= tau' => R(jj', i') <= R(j', i')
IH: forall (tau'0,tau:timestamp),
tau'0 < tau' =>
happens(tau, tau'0) => i <> i' => sT i@tau <> sR i'@tau'0
Meq: sT i@tau = sR i'@tau'
----------------------------------------
false



+ rewrite Meq B /sT in A0.
[> Line 408: (rewrite) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,i',j,j':index[const],tau,tau':timestamp[const]
A0: sR i'@R(j', i') = H (sT i@pred (T(i, j)), k)
A1: T(i, j) <= tau
B: sR i'@tau' = sR i'@R(j', i')
C: R(j', i') <= tau'
D: happens(tau, tau')
E: i <> i'
Hsup: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Hsup': forall (jj':index),
happens(R(jj', i')) && R(jj', i') <= tau' => R(jj', i') <= R(j', i')
IH: forall (tau'0,tau:timestamp),
tau'0 < tau' =>
happens(tau, tau'0) => i <> i' => sT i@tau <> sR i'@tau'0
Meq: sT i@tau = sR i'@tau'
----------------------------------------
false


expand sR(i')@R(j',i').
[> Line 409: (expand) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,i',j,j':index[const],tau,tau':timestamp[const]
A0: H (sR i'@pred (R(j', i')), k) = H (sT i@pred (T(i, j)), k)
A1: T(i, j) <= tau
B: sR i'@tau' = H (sR i'@pred (R(j', i')), k)
C: R(j', i') <= tau'
D: happens(tau, tau')
E: i <> i'
Hsup: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Hsup': forall (jj':index),
happens(R(jj', i')) && R(jj', i') <= tau' => R(jj', i') <= R(j', i')
IH: forall (tau'0,tau:timestamp),
tau'0 < tau' =>
happens(tau, tau'0) => i <> i' => sT i@tau <> sR i'@tau'0
Meq: sT i@tau = sR i'@tau'
----------------------------------------
false


collision A0 => H.
[> Line 410: ((collision);(intro)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,i',j,j':index[const],tau,tau':timestamp[const]
A0: H (sR i'@pred (R(j', i')), k) = H (sT i@pred (T(i, j)), k)
A1: T(i, j) <= tau
B: sR i'@tau' = H (sR i'@pred (R(j', i')), k)
C: R(j', i') <= tau'
D: happens(tau, tau')
E: i <> i'
H: sR i'@pred (R(j', i')) = sT i@pred (T(i, j))
Hsup: forall (j':index),
happens(T(i, j')) && T(i, j') <= tau => T(i, j') <= T(i, j)
Hsup': forall (jj':index),
happens(R(jj', i')) && R(jj', i') <= tau' => R(jj', i') <= R(j', i')
IH: forall (tau'0,tau:timestamp),
tau'0 < tau' =>
happens(tau, tau'0) => i <> i' => sT i@tau <> sR i'@tau'0
Meq: sT i@tau = sR i'@tau'
----------------------------------------
false


by use IH with pred(R(j',i')),pred(T(i,j)).
[> Line 411: by (have) [goal> lemma disjoint_chains is proved


Qed.
lemma disjoint_chains @system:(set:default; equiv:None) :
forall (tau',tau:timestamp,i',i:index),
happens(tau, tau') => i <> i' => sT i@tau <> sR i'@tau'
Exiting proof mode.



(* SECURITY PROPERTIES *)

lemma authentication (jj,ii:index):
happens(R(jj,ii)) =>
cond@R(jj,ii) =>
exists (j:index), T(ii,j) < R(jj,ii) && output@T(ii,j) = input@R(jj,ii).
Goal authentication :
happens(R(jj, ii)) =>
cond@R(jj, ii) =>
exists (j:index), T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)

Proof.
[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index
----------------------------------------
happens(R(jj, ii)) =>
cond@R(jj, ii) =>
exists (j:index), T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)


intro Hap @/cond Hcond.
[> Line 421: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const]
Hap: happens(R(jj, ii))
Hcond: input@R(jj, ii) = G (H (sR ii@pred (R(jj, ii)), k), k')
----------------------------------------
exists (j:index), T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)


euf Hcond.
[> Line 422: (euf) Indirect bad occurrences of key k', and messages authenticated by it
in other actions:
sT i@T(i, j) auth. by k'
(collision with H (sR ii@pred (R(jj, ii)), k) auth. by k')
in action T(i, j)
in term (happens(T(i, j)), G (sT i@T(i, j), k'))

Total: 1 occurrence
0 of them are subsumed by another
1 occurrence remaining

[goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: ii,jj:index[const]
Hap: happens(R(jj, ii))
Hcond: input@R(jj, ii) = G (H (sR ii@pred (R(jj, ii)), k), k')
----------------------------------------
(exists (i,j:index),
T(i, j) < R(jj, ii) && H (sR ii@pred (R(jj, ii)), k) = sT i@T(i, j))
=>
exists (j:index), T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)


intro [i j [Ht M]].
[> Line 423: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
Hap: happens(R(jj, ii))
Hcond: input@R(jj, ii) = G (H (sR ii@pred (R(jj, ii)), k), k')
Ht: T(i, j) < R(jj, ii)
M: H (sR ii@pred (R(jj, ii)), k) = sT i@T(i, j)
----------------------------------------
exists (j:index), T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)


case (i=ii) => _; 1: by exists j.
[> Line 424: (((case);(intro)); 1: by (exists)) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
Hap: happens(R(jj, ii))
Hcond: input@R(jj, ii) = G (H (sR ii@pred (R(jj, ii)), k), k')
Ht: T(i, j) < R(jj, ii)
M: H (sR ii@pred (R(jj, ii)), k) = sT i@T(i, j)
_: not (i = ii)
----------------------------------------
exists (j:index), T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)


rewrite /sT in M.
[> Line 425: (rewrite) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
Hap: happens(R(jj, ii))
Hcond: input@R(jj, ii) = G (H (sR ii@pred (R(jj, ii)), k), k')
Ht: T(i, j) < R(jj, ii)
M: H (sR ii@pred (R(jj, ii)), k) = H (sT i@pred (T(i, j)), k)
_: not (i = ii)
----------------------------------------
exists (j:index), T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)


collision.
[> Line 426: (collision) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
Hap: happens(R(jj, ii))
Hcond: input@R(jj, ii) = G (H (sR ii@pred (R(jj, ii)), k), k')
Ht: T(i, j) < R(jj, ii)
M: H (sR ii@pred (R(jj, ii)), k) = H (sT i@pred (T(i, j)), k)
_: not (i = ii)
----------------------------------------
sR ii@pred (R(jj, ii)) = sT i@pred (T(i, j)) =>
exists (j:index), T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)


intro Meq.
[> Line 427: (intro) [goal> Focused goal (1/1):
System: (set:default; equiv:None)
Variables: i,ii,j,jj:index[const]
Hap: happens(R(jj, ii))
Hcond: input@R(jj, ii) = G (H (sR ii@pred (R(jj, ii)), k), k')
Ht: T(i, j) < R(jj, ii)
M: H (sR ii@pred (R(jj, ii)), k) = H (sT i@pred (T(i, j)), k)
Meq: sR ii@pred (R(jj, ii)) = sT i@pred (T(i, j))
_: not (i = ii)
----------------------------------------
exists (j:index), T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)


by use disjoint_chains with pred(R(jj,ii)),pred(T(i,j)),ii,i.
[> Line 428: by (have) [goal> lemma authentication is proved


Qed.
lemma authentication @system:(set:default; equiv:None) :
forall (jj,ii:index),
happens(R(jj, ii)) =>
cond@R(jj, ii) =>
exists (j:index),
T(ii, j) < R(jj, ii) && output@T(ii, j) = input@R(jj, ii)
Exiting proof mode.

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