include Basic. Goal eq_sym :
(x = y) = (y = x)
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x,y:'a
----------------------------------------
(x = y) = (y = x)

[> Line 11: by (rewrite ...)
[goal> Goal eq_sym is proved
Exiting proof mode.


Goal neq_sym :
(x <> y) = (y <> x)
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x,y:'a
----------------------------------------
(x <> y) = (y <> x)

[> Line 14: by (rewrite ...)
[goal> Goal neq_sym is proved
Exiting proof mode.


Goal eq_refl :
(x = x) = true
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x:'a
----------------------------------------
(x = x) = true

[> Line 18: by (rewrite ...)
[goal> Goal eq_refl is proved
Exiting proof mode.


Goal false_true :
(false = true) = false
[goal> Focused goal (1/1):
System: any
----------------------------------------
(false = true) = false

[> Line 30: by (rewrite ...)
[goal> Goal false_true is proved
Exiting proof mode.


Goal eq_true :
(b = true) = b
[goal> Focused goal (1/1):
System: any
Variables: b:bool
----------------------------------------
(b = true) = b

[> Line 35: by (case b)
[goal> Goal eq_true is proved
Exiting proof mode.


Goal eq_true2 :
(true = b) = b
[goal> Focused goal (1/1):
System: any
Variables: b:bool
----------------------------------------
(true = b) = b

[> Line 39: by (case b)
[goal> Goal eq_true2 is proved
Exiting proof mode.


Goal not_not :
not(not(b)) = b
[goal> Focused goal (1/1):
System: any
Variables: b:bool
----------------------------------------
not(not(b)) = b

[> Line 54: by (case b)
[goal> Goal not_not is proved
Exiting proof mode.


Goal not_eq :
not(x = y) = (x <> y)
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x,y:'a
----------------------------------------
not(x = y) = (x <> y)

[> Line 60: by (rewrite ...)
[goal> Goal not_eq is proved
Exiting proof mode.


Goal not_neq :
not(x <> y) = (x = y)
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x,y:'a
----------------------------------------
not(x <> y) = (x = y)

[> Line 66: by (rewrite ...)
[goal> Goal not_neq is proved
Exiting proof mode.


Goal not_eqfalse :
(b = false) = not(b)
[goal> Focused goal (1/1):
System: any
Variables: b:bool
----------------------------------------
(b = false) = not(b)

[> Line 72: by (case b)
[goal> Goal not_eqfalse is proved
Exiting proof mode.


Goal eq_false :
((x = y) = false) = (x <> y)
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x,y:'a
----------------------------------------
((x = y) = false) = (x <> y)

[> Line 80: (rewrite ...)
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x,y:'a
----------------------------------------
((x = y) = false) = not(x = y)

[> Line 80: ((case (x = y));(intro _))
[goal> Focused goal (1/2):
System: any
Type variables: 'a
Variables: x,y:'a
_: x = y
----------------------------------------
(true = false) = not(true)

[> Line 80: simpl
[goal> Focused goal (1/2):
System: any
Type variables: 'a
Variables: x,y:'a
_: x = y
----------------------------------------
true

[> Line 80: auto
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x,y:'a
_: not(x = y)
----------------------------------------
(false = false) = not(false)

[> Line 81: by (rewrite ...)
[goal> Goal eq_false is proved
Exiting proof mode.


Goal and_true_r :
(b && true) = b
[goal> Focused goal (1/1):
System: any
Variables: b:bool
----------------------------------------
(b && true) = b

[> Line 94: by (rewrite ... ...)
[goal> Goal and_true_r is proved
Exiting proof mode.


Goal and_false_r :
(b && false) = false
[goal> Focused goal (1/1):
System: any
Variables: b:bool
----------------------------------------
(b && false) = false

[> Line 101: by (rewrite ... ...)
[goal> Goal and_false_r is proved
Exiting proof mode.


Goal or_false_r :
(b || false) = b
[goal> Focused goal (1/1):
System: any
Variables: b:bool
----------------------------------------
(b || false) = b

[> Line 112: by (rewrite ... ...)
[goal> Goal or_false_r is proved
Exiting proof mode.


Goal or_true_r :
(b || true) = true
[goal> Focused goal (1/1):
System: any
Variables: b:bool
----------------------------------------
(b || true) = true

[> Line 119: by (rewrite ... ...)
[goal> Goal or_true_r is proved
Exiting proof mode.


Goal not_and :
not((a && b)) = (not(a) || not(b))
[goal> Focused goal (1/1):
System: any
Variables: a,b:bool
----------------------------------------
not((a && b)) = (not(a) || not(b))

[> Line 128: (rewrite ...)
[goal> Focused goal (1/1):
System: any
Variables: a,b:bool
----------------------------------------
not((a && b)) <=> not(a) || not(b)

[> Line 129: (((case a);(case b));(intro //=))
[goal> Goal not_and is proved
Exiting proof mode.


Goal not_or :
not((a || b)) = (not(a) && not(b))
[goal> Focused goal (1/1):
System: any
Variables: a,b:bool
----------------------------------------
not((a || b)) = (not(a) && not(b))

[> Line 134: (rewrite ...)
[goal> Focused goal (1/1):
System: any
Variables: a,b:bool
----------------------------------------
not((a || b)) <=> not(a) && not(b)

[> Line 135: (((case a);(case b));(intro //=))
[goal> Goal not_or is proved
Exiting proof mode.


Goal if_true :
b => if b then x else y = x
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: b:bool,x,y:'a
----------------------------------------
b => if b then x else y = x

[> Line 144: (intro *)
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: b:bool,x,y:'a
H: b
----------------------------------------
if b then x else y = x

[> Line 145: (case if b then x else y)
[goal> Focused goal (1/2):
System: any
Type variables: 'a
Variables: b:bool,x,y:'a
H: b
----------------------------------------
b && if b then x else y = x => x = x

[> Line 146: auto
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: b:bool,x,y:'a
H: b
----------------------------------------
not(b) && if b then x else y = y => y = x

[> Line 147: (intro [HH _])
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: b:bool,x,y:'a
H: b
HH: not(b)
_: if b then x else y = y
----------------------------------------
y = x

[> Line 147: by (have ... as )
[goal> Goal if_true is proved
Exiting proof mode.


Goal if_true0 :
if true then x else y = x
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x,y:'a
----------------------------------------
if true then x else y = x

[> Line 153: by (rewrite ...)
[goal> Goal if_true0 is proved
Exiting proof mode.


Goal if_false :
not(b) => if b then x else y = y
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: b:bool,x,y:'a
----------------------------------------
not(b) => if b then x else y = y

[> Line 160: ((intro *);(case if b then x else y))
[goal> Focused goal (1/2):
System: any
Type variables: 'a
Variables: b:bool,x,y:'a
H: not(b)
----------------------------------------
b && if b then x else y = x => x = y

[> Line 161: (intro [HH _])
[goal> Focused goal (1/2):
System: any
Type variables: 'a
Variables: b:bool,x,y:'a
H: not(b)
HH: b
_: if b then x else y = x
----------------------------------------
x = y

[> Line 161: by (have ... as )
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: b:bool,x,y:'a
H: not(b)
----------------------------------------
not(b) && if b then x else y = y => y = y

[> Line 162: auto
[goal> Goal if_false is proved
Exiting proof mode.


Goal if_false0 :
if false then x else y = y
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: x,y:'a
----------------------------------------
if false then x else y = y

[> Line 168: by (rewrite ...)
[goal> Goal if_false0 is proved
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: any
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 175: by ((case b);(case b'))
[goal> Goal if_then_then is proved
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: any
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 182: ((case b);
((intro H);((case b');((intro H');(simpl;(try auto))))))

[goal> Focused goal (1/2):
System: any
Type variables: 'a
Variables: b,b':bool,x,y,z:'a
H: b
H': b'
----------------------------------------
x = if (true => true) then x else y

[> Line 183: by (rewrite ...)
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: b,b':bool,x,y,z:'a
H: b
H': not(b')
----------------------------------------
y = if (true => false) then x else y

[> Line 184: (rewrite ...)
[goal> Focused goal (1/2):
System: any
Type variables: 'a
Variables: b,b':bool,x,y,z:'a
H: b
H': not(b')
----------------------------------------
not((true => false))

[> Line 185: ((intro Habs);by (have ... as ))
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: b,b':bool,x,y,z:'a
H: b
H': not(b')
----------------------------------------
y = y

[> Line 186: auto
[goal> Goal if_then_implies is proved
Exiting proof mode.


Goal if_same :
if b then x else x = x
[goal> Focused goal (1/1):
System: any
Type variables: 'a
Variables: b:bool,x:'a
----------------------------------------
if b then x else x = x

[> Line 192: by (case b)
[goal> Goal if_same is proved
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: any
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 201: by ((intro ->);(case b'))
[goal> Goal if_then is proved
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: any
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 210: by ((intro ->);(case b'))
[goal> Goal if_else is proved
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: any
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 219: by ((intro ->);(case b'))
[goal> Goal if_then_not is proved
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: any
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 228: by ((intro ->);(case b'))
[goal> Goal if_else_not is proved
Exiting proof mode.


Goal fst_pair :
fst(<x,y>) = x
[goal> Focused goal (1/1):
System: any
Variables: x,y:message
----------------------------------------
fst(<x,y>) = x

[> Line 236: auto
[goal> Goal fst_pair is proved
Exiting proof mode.


Goal snd_pair :
snd(<x,y>) = y
[goal> Focused goal (1/1):
System: any
Variables: x,y:message
----------------------------------------
snd(<x,y>) = y

[> Line 240: auto
[goal> Goal snd_pair is proved
Exiting proof mode.


Goal iff_refl :
(x <=> x) = true
[goal> Focused goal (1/1):
System: any
Variables: x:bool
----------------------------------------
(x <=> x) = true

[> Line 248: by (rewrite ...)
[goal> Goal iff_refl is proved
Exiting proof mode.


Goal iff_sym :
(x <=> y) = (y <=> x)
[goal> Focused goal (1/1):
System: any
Variables: x,y:bool
----------------------------------------
(x <=> y) = (y <=> x)

[> Line 254: by (rewrite ...)
[goal> Goal iff_sym is proved
Exiting proof mode.


Goal true_iff_false :
(true <=> false) = false
[goal> Focused goal (1/1):
System: any
----------------------------------------
(true <=> false) = false

[> Line 259: by (rewrite ...)
[goal> Goal true_iff_false is proved
Exiting proof mode.


Goal false_iff_true :
(false <=> true) = false
[goal> Focused goal (1/1):
System: any
----------------------------------------
(false <=> true) = false

[> Line 265: by (rewrite ...)
[goal> Goal false_iff_true is proved
Exiting proof mode.


Goal exists_false1 :
(exists (a:'a), false) = false
[goal> Focused goal (1/1):
System: any
Type variables: 'a
----------------------------------------
(exists (a:'a), false) = false

[> Line 277: by (rewrite ...)
[goal> Goal exists_false1 is proved
Exiting proof mode.


Goal exists_false2 :
(exists (a:'a,b:'b), false) = false
[goal> Focused goal (1/1):
System: any
Type variables: 'a, 'b
----------------------------------------
(exists (a:'a,b:'b), false) = false

[> Line 281: by (rewrite ...)
[goal> Goal exists_false2 is proved
Exiting proof mode.


Goal exists_false3 :
(exists (a:'a,b:'b,c:'c), false) = false
[goal> Focused goal (1/1):
System: any
Type variables: 'a, 'b, 'c
----------------------------------------
(exists (a:'a,b:'b,c:'c), false) = false

[> Line 285: by (rewrite ...)
[goal> Goal exists_false3 is proved
Exiting proof mode.


Goal exists_false4 :
(exists (a:'a,b:'b,c:'c,d:'d), false) = false
[goal> Focused goal (1/1):
System: any
Type variables: 'a, 'b, 'c, 'd
----------------------------------------
(exists (a:'a,b:'b,c:'c,d:'d), false) = false

[> Line 289: by (rewrite ...)
[goal> Goal exists_false4 is proved
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: any
Type variables: 'a, 'b, 'c, 'd, 'e
----------------------------------------
(exists (a:'a,b:'b,c:'c,d:'d,e:'e), false) = false

[> Line 293: by (rewrite ...)
[goal> Goal exists_false5 is proved
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: any
Type variables: 'a, 'b, 'c, 'd, 'e, 'f
----------------------------------------
(exists (a:'a,b:'b,c:'c,d:'d,e:'e,f:'f), false) = false

[> Line 297: by (rewrite ...)
[goal> Goal exists_false6 is proved
Exiting proof mode.


Goal forall_true1 :
(forall (a:'a), true) = true
[goal> Focused goal (1/1):
System: any
Type variables: 'a
----------------------------------------
(forall (a:'a), true) = true

[> Line 307: auto
[goal> Goal forall_true1 is proved
Exiting proof mode.


Goal forall_true2 :
(forall (a:'a,b:'b), true) = true
[goal> Focused goal (1/1):
System: any
Type variables: 'a, 'b
----------------------------------------
(forall (a:'a,b:'b), true) = true

[> Line 311: auto
[goal> Goal forall_true2 is proved
Exiting proof mode.


Goal forall_true3 :
(forall (a:'a,b:'b,c:'c), true) = true
[goal> Focused goal (1/1):
System: any
Type variables: 'a, 'b, 'c
----------------------------------------
(forall (a:'a,b:'b,c:'c), true) = true

[> Line 315: auto
[goal> Goal forall_true3 is proved
Exiting proof mode.


Goal forall_true4 :
(forall (a:'a,b:'b,c:'c,d:'d), true) = true
[goal> Focused goal (1/1):
System: any
Type variables: 'a, 'b, 'c, 'd
----------------------------------------
(forall (a:'a,b:'b,c:'c,d:'d), true) = true

[> Line 319: auto
[goal> Goal forall_true4 is proved
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: any
Type variables: 'a, 'b, 'c, 'd, 'e
----------------------------------------
(forall (a:'a,b:'b,c:'c,d:'d,e:'e), true) = true

[> Line 323: auto
[goal> Goal forall_true5 is proved
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: any
Type variables: 'a, 'b, 'c, 'd, 'e, 'f
----------------------------------------
(forall (a:'a,b:'b,c:'c,d:'d,e:'e,f:'f), true) = true

[> Line 327: auto
[goal> Goal forall_true6 is proved
Exiting proof mode.


[warning> loaded: Basic.sp <]

BASIC HASH

The Basic Hash protocol as described in [A] is an RFID protocol involving:

  • a tag associated to a secret key;
  • generic readers having access to a database containing all these keys.

The protocol is as follows:

T --> R : <nT, h(nT,key)>
R --> T : ok

In this file, we prove an authentication property for the reader.

[A] Mayla Brusò, Kostas Chatzikokolakis, and Jerry den Hartog. Formal Verification of Privacy for RFID Systems. pages 75–88, July 2010.

Include basic standard library.

 hash h

abstract ok : message
abstract ko : message.

We start by declaring the function symbol h for the hash function, as well as two public constants ok and ko (used by the reader).

 name key : index -> message.

Name for modelling the keys of tags.

 channel cT
channel cR.

Finally, we declare the channels used by the protocol.

 process tag(i:index,k:index) =
new nT;
out(cT, <nT, h(nT,key(i))>).

Session k of tag i.

 process reader(j:index) =
in(cT,x);
try find i such that snd(x) = h(fst(x),key(i)) in
out(cR,ok)
else
out(cR,ko).

Session j of reader.

system [BasicHash] ((!_j R: reader(j)) | (!_i !_k T: tag(i,k))). System before processing:

(!_j R: reader j) | (!_i !_k T: tag i k)

System after processing:

(!_j
in(cT,x);
find (i) such that (snd(x) = h(fst(x),key(i))) in R: out(cR,ok); null
else R1: out(cR,ko); null) |
(!_i !_k T: out(cT,pair(nT(i,k),h(nT(i,k),key(i)))); null)

System BasicHash registered with actions (init,R,R1,T).
 print system [BasicHash]. System [left:BasicHash/left, right:BasicHash/right]
Available actions:

action name: init
condition: true
output: empty

action name: R(j,i)
indices: j,i
condition: snd(input@R(j,i)) = h(fst(input@R(j,i)),key(i))
output: ok

action name: R1(j)
indices: j
condition:
forall (i:index), not(snd(input@R1(j)) = h(fst(input@R1(j)),key(i)))
output: ko

action name: T(i,k)
indices: i,k
condition: true
output: <nT(i,k),h(nT(i,k),key(i))>

Show the set of actions obtained from above process.

 goal [BasicHash] wa_R :
forall (tau:timestamp),
happens(tau) =>
((exists (i:index),
snd(input@tau) = h(fst(input@tau),key(i)))
<=>
(exists (i,k:index), T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) &&
snd(output@T(i,k)) = snd(input@tau))). Goal wa_R :
forall (tau:timestamp),
happens(tau) =>
(exists (i:index), snd(input@tau) = h(fst(input@tau),key(i))) <=>
exists (i,k:index),
T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) &&
snd(output@T(i,k)) = snd(input@tau)

Whenever a reader accepts a message (i.e. the condition of the action R(j) evaluates to true), there exists an action T(i,k) that has been executed before the reader, and such that the input of the reader corresponds to the output of this tag (and conversely).

The same holds for R1 (the else branch of the reader) but with a negation. We will prove once and for all a property that is generalized for any tau, which will be useful later for tau = R(j) and tau = R1(j).

 Proof. [goal> Focused goal (1/1):
System: left:BasicHash/left, right:BasicHash/right
----------------------------------------
forall (tau:timestamp),
happens(tau) =>
(exists (i:index), snd(input@tau) = h(fst(input@tau),key(i))) <=>
exists (i,k:index),
T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) &&
snd(output@T(i,k)) = snd(input@tau)

The high-level idea of the proof is to use the EUF cryptographic axiom: only the tag T(i,k) can forge h(nT(i,k),key(i)) because the secret key is not known by the attacker. Therefore, any message accepted by the reader must come from a tag that has played before. The converse implication is trivial because any honest tag output is accepted by the reader.
intro tau Hap. [> Line 87: (intro tau Hap)
[goal> Focused goal (1/1):
System: left:BasicHash/left, right:BasicHash/right
Variables: tau:timestamp
Hap: happens(tau)
----------------------------------------
(exists (i:index), snd(input@tau) = h(fst(input@tau),key(i))) <=>
exists (i,k:index),
T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) && snd(output@T(i,k)) = snd(input@tau)

 split. [> Line 87: split
[goal> Focused goal (1/2):
System: left:BasicHash/left, right:BasicHash/right
Variables: tau:timestamp
Hap: happens(tau)
----------------------------------------
(exists (i:index), snd(input@tau) = h(fst(input@tau),key(i))) =>
exists (i,k:index),
T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) && snd(output@T(i,k)) = snd(input@tau)
+ intro [i Meq]. [> Line 88: (intro [i Meq])
[goal> Focused goal (1/2):
System: left:BasicHash/left, right:BasicHash/right
Variables: i:index,tau:timestamp
Hap: happens(tau)
Meq: snd(input@tau) = h(fst(input@tau),key(i))
----------------------------------------
exists (i,k:index),
T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) && snd(output@T(i,k)) = snd(input@tau)

 euf Meq => *. [> Line 94: ((euf Meq);(intro *))
[goal> Focused goal (1/2):
System: left:BasicHash/left, right:BasicHash/right
Variables: i,k:index,tau:timestamp
Clt: T(i,k) < tau
Hap: happens(tau)
Meq: snd(input@tau) = h(fst(input@tau),key(i))
Meq0: nT(i,k) = fst(input@tau)
----------------------------------------
exists (i,k:index),
T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) && snd(output@T(i,k)) = snd(input@tau)

Applying the euf tactic on the Meq hypothesis generates a new hypothesis stating that fst(input@R(j)) must be equal to some message that has already been hashed before. The only possibility is that this hash comes from the output of a tag that has played before (thus the new hypothesis on timestamps).

 exists i,k. [> Line 94: (exists i, k)
[goal> Focused goal (1/2):
System: left:BasicHash/left, right:BasicHash/right
Variables: i,k:index,tau:timestamp
Clt: T(i,k) < tau
Hap: happens(tau)
Meq: snd(input@tau) = h(fst(input@tau),key(i))
Meq0: nT(i,k) = fst(input@tau)
----------------------------------------
T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) && snd(output@T(i,k)) = snd(input@tau)

 auto. [> Line 94: auto
[goal> Focused goal (1/1):
System: left:BasicHash/left, right:BasicHash/right
Variables: tau:timestamp
Hap: happens(tau)
----------------------------------------
(exists (i,k:index),
T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) && snd(output@T(i,k)) = snd(input@tau)) =>
exists (i:index), snd(input@tau) = h(fst(input@tau),key(i))

 + intro [i k Meq]. [> Line 97: (intro [i k Meq])
[goal> Focused goal (1/1):
System: left:BasicHash/left, right:BasicHash/right
Variables: i,k:index,tau:timestamp
Hap: happens(tau)
Meq: T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) &&
snd(output@T(i,k)) = snd(input@tau)
----------------------------------------
exists (i:index), snd(input@tau) = h(fst(input@tau),key(i))

For the second implication (<=), the conclusion of the goal can directly be obtained from the hypotheses.

 exists i. [> Line 97: (exists i)
[goal> Focused goal (1/1):
System: left:BasicHash/left, right:BasicHash/right
Variables: i,k:index,tau:timestamp
Hap: happens(tau)
Meq: T(i,k) < tau &&
fst(output@T(i,k)) = fst(input@tau) &&
snd(output@T(i,k)) = snd(input@tau)
----------------------------------------
snd(input@tau) = h(fst(input@tau),key(i))

 auto. [> Line 97: auto
[goal> Goal wa_R is proved

Qed. Exiting proof mode.

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