Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
Datasets:
fact stringlengths 1 199k | statement stringlengths 1 33.2k | proof stringlengths 0 198k | type stringclasses 14
values | kind stringclasses 4
values | symbolic_name stringlengths 1 92 | library stringclasses 350
values | filename stringlengths 17 103 | imports listlengths 0 0 | deps listlengths 0 64 | docstring stringclasses 0
values | line_start int64 1 25k | line_end int64 1 25k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value | content_level stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
`(~p = F) = p | `(~p = F) = p | prove | theorem | NEG_EQ_F | examples | examples/dpll.sml | [] | [] | null | 105 | 105 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
Definition divides_def:
a divides b <=> ?x. b = a * x
End | Definition divides_def:
a divides b <=> ?x. b = a * x
End | Definition | definition | divides_def | examples | examples/euclid.sml | [] | [
"x"
] | null | 23 | 25 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
Definition prime_def:
prime p <=> p <> 1 /\ !x. x divides p ==> (x = 1) \/ (x = p)
End | Definition prime_def:
prime p <=> p <> 1 /\ !x. x divides p ==> (x = 1) \/ (x = p)
End | Definition | definition | prime_def | examples | examples/euclid.sml | [] | [
"x"
] | null | 31 | 33 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
Theorem DIVIDES_ZERO:
!x. x divides 0
Proof
metis_tac [divides_def,MULT_CLAUSES]
QED | Theorem DIVIDES_ZERO:
!x. x divides 0 | Proof
metis_tac [divides_def,MULT_CLAUSES]
QED | Theorem | theorem | DIVIDES_ZERO | examples | examples/euclid.sml | [] | [
"MULT_CLAUSES",
"divides_def",
"x"
] | null | 39 | 43 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem ZERO_DIVIDES:
!x. 0 divides x <=> x = 0
Proof
metis_tac [divides_def,MULT_CLAUSES]
QED | Theorem ZERO_DIVIDES:
!x. 0 divides x <=> x = 0 | Proof
metis_tac [divides_def,MULT_CLAUSES]
QED | Theorem | theorem | ZERO_DIVIDES | examples | examples/euclid.sml | [] | [
"MULT_CLAUSES",
"divides_def",
"x"
] | null | 45 | 49 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_ONE:
!x. x divides 1 <=> x = 1
Proof
metis_tac [divides_def,MULT_CLAUSES,MULT_EQ_1]
QED | Theorem DIVIDES_ONE:
!x. x divides 1 <=> x = 1 | Proof
metis_tac [divides_def,MULT_CLAUSES,MULT_EQ_1]
QED | Theorem | theorem | DIVIDES_ONE | examples | examples/euclid.sml | [] | [
"MULT_CLAUSES",
"MULT_EQ_1",
"divides_def",
"x"
] | null | 51 | 55 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_REFL :
!x. x divides x
Proof
metis_tac [divides_def,MULT_CLAUSES]
QED | Theorem DIVIDES_REFL :
!x. x divides x | Proof
metis_tac [divides_def,MULT_CLAUSES]
QED | Theorem | theorem | DIVIDES_REFL | examples | examples/euclid.sml | [] | [
"MULT_CLAUSES",
"divides_def",
"x"
] | null | 57 | 61 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_TRANS :
!a b c. a divides b /\ b divides c ==> a divides c
Proof
metis_tac [divides_def,MULT_ASSOC]
QED | Theorem DIVIDES_TRANS :
!a b c. a divides b /\ b divides c ==> a divides c | Proof
metis_tac [divides_def,MULT_ASSOC]
QED | Theorem | theorem | DIVIDES_TRANS | examples | examples/euclid.sml | [] | [
"MULT_ASSOC",
"divides_def"
] | null | 63 | 67 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_ADD :
!d a b. d divides a /\ d divides b ==> d divides (a + b)
Proof
metis_tac[divides_def,LEFT_ADD_DISTRIB]
QED | Theorem DIVIDES_ADD :
!d a b. d divides a /\ d divides b ==> d divides (a + b) | Proof
metis_tac[divides_def,LEFT_ADD_DISTRIB]
QED | Theorem | theorem | DIVIDES_ADD | examples | examples/euclid.sml | [] | [
"LEFT_ADD_DISTRIB",
"divides_def"
] | null | 69 | 73 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_SUB :
!d a b. d divides a /\ d divides b ==> d divides (a - b)
Proof
metis_tac [divides_def,LEFT_SUB_DISTRIB]
QED | Theorem DIVIDES_SUB :
!d a b. d divides a /\ d divides b ==> d divides (a - b) | Proof
metis_tac [divides_def,LEFT_SUB_DISTRIB]
QED | Theorem | theorem | DIVIDES_SUB | examples | examples/euclid.sml | [] | [
"LEFT_SUB_DISTRIB",
"divides_def"
] | null | 75 | 79 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_ADDL :
!d a b. d divides a /\ d divides (a + b) ==> d divides b
Proof
metis_tac [ADD_SUB,ADD_SYM,DIVIDES_SUB]
QED | Theorem DIVIDES_ADDL :
!d a b. d divides a /\ d divides (a + b) ==> d divides b | Proof
metis_tac [ADD_SUB,ADD_SYM,DIVIDES_SUB]
QED | Theorem | theorem | DIVIDES_ADDL | examples | examples/euclid.sml | [] | [
"ADD_SUB",
"ADD_SYM",
"DIVIDES_SUB"
] | null | 81 | 85 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_LMUL :
!d a x. d divides a ==> d divides (x * a)
Proof
metis_tac [divides_def,MULT_ASSOC,MULT_SYM]
QED | Theorem DIVIDES_LMUL :
!d a x. d divides a ==> d divides (x * a) | Proof
metis_tac [divides_def,MULT_ASSOC,MULT_SYM]
QED | Theorem | theorem | DIVIDES_LMUL | examples | examples/euclid.sml | [] | [
"MULT_ASSOC",
"MULT_SYM",
"divides_def",
"x"
] | null | 87 | 91 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_RMUL :
!d a x. d divides a ==> d divides (a * x)
Proof
metis_tac [MULT_SYM,DIVIDES_LMUL]
QED | Theorem DIVIDES_RMUL :
!d a x. d divides a ==> d divides (a * x) | Proof
metis_tac [MULT_SYM,DIVIDES_LMUL]
QED | Theorem | theorem | DIVIDES_RMUL | examples | examples/euclid.sml | [] | [
"DIVIDES_LMUL",
"MULT_SYM",
"x"
] | null | 93 | 97 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_LE :
!a b. a divides b ==> (0 < a ∧ a <= b) \/ b = 0
Proof
rw [divides_def] >> rw[]
QED | Theorem DIVIDES_LE :
!a b. a divides b ==> (0 < a ∧ a <= b) \/ b = 0 | Proof
rw [divides_def] >> rw[]
QED | Theorem | theorem | DIVIDES_LE | examples | examples/euclid.sml | [] | [
"divides_def",
"rw"
] | null | 99 | 103 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n)
Proof
rw [LESS_EQ_EXISTS]
>> Induct_on `p`
>> rw [FACT,ADD_CLAUSES]
>> Cases_on `m`
>> metis_tac [FACT, DECIDE ``!x. ~(x < x)``,
DIVIDES_RMUL, DIVIDES_LMUL, DIVIDES_REFL]
QED | Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n) | Proof
rw [LESS_EQ_EXISTS]
>> Induct_on `p`
>> rw [FACT,ADD_CLAUSES]
>> Cases_on `m`
>> metis_tac [FACT, DECIDE ``!x. ~(x < x)``,
DIVIDES_RMUL, DIVIDES_LMUL, DIVIDES_REFL]
QED | Theorem | theorem | LE_DIVIDES_FACT | examples | examples/euclid.sml | [] | [
"ADD_CLAUSES",
"DIVIDES_LMUL",
"DIVIDES_REFL",
"DIVIDES_RMUL",
"FACT",
"LESS_EQ_EXISTS",
"rw",
"x"
] | null | 109 | 118 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem DIVIDES_FACT:
∀n. 0 < n ==> n divides (FACT n)
Proof
Cases
>> rw[FACT]
>> rename1 ‘SUC n’
>> irule DIVIDES_LMUL
>> metis_tac [DIVIDES_REFL]
QED | Theorem DIVIDES_FACT:
∀n. 0 < n ==> n divides (FACT n) | Proof
Cases
>> rw[FACT]
>> rename1 ‘SUC n’
>> irule DIVIDES_LMUL
>> metis_tac [DIVIDES_REFL]
QED | Theorem | theorem | DIVIDES_FACT | examples | examples/euclid.sml | [] | [
"DIVIDES_LMUL",
"DIVIDES_REFL",
"FACT",
"rw"
] | null | 120 | 128 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n)
Proof
rw [LESS_EQ_EXISTS]
>> Induct_on `p`
>> fs [FACT,ADD_CLAUSES]
>- metis_tac [DIVIDES_FACT]
>- metis_tac [DIVIDES_RMUL]
QED | Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n) | Proof
rw [LESS_EQ_EXISTS]
>> Induct_on `p`
>> fs [FACT,ADD_CLAUSES]
>- metis_tac [DIVIDES_FACT]
>- metis_tac [DIVIDES_RMUL]
QED | Theorem | theorem | LE_DIVIDES_FACT | examples | examples/euclid.sml | [] | [
"ADD_CLAUSES",
"DIVIDES_FACT",
"DIVIDES_RMUL",
"FACT",
"LESS_EQ_EXISTS",
"rw"
] | null | 130 | 138 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n)
Proof
`!m p. 0 < m ==> m divides FACT (m + p)`
suffices_by metis_tac[LESS_EQ_EXISTS]
>> Induct_on `p`
>> rw [FACT,ADD_CLAUSES]
>> metis_tac [FACT, DECIDE ``!x. ~(x < x)``, num_CASES,
DIVIDES_RMUL, DIVIDES_LMUL, DIV... | Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n) | Proof
`!m p. 0 < m ==> m divides FACT (m + p)`
suffices_by metis_tac[LESS_EQ_EXISTS]
>> Induct_on `p`
>> rw [FACT,ADD_CLAUSES]
>> metis_tac [FACT, DECIDE ``!x. ~(x < x)``, num_CASES,
DIVIDES_RMUL, DIVIDES_LMUL, DIVIDES_REFL]
QED | Theorem | theorem | LE_DIVIDES_FACT | examples | examples/euclid.sml | [] | [
"ADD_CLAUSES",
"DIVIDES_LMUL",
"DIVIDES_REFL",
"DIVIDES_RMUL",
"FACT",
"LESS_EQ_EXISTS",
"num_CASES",
"rw",
"x"
] | null | 140 | 149 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n)
Proof
rw [LESS_EQ_EXISTS]
>> Induct_on `p`
>> metis_tac [FACT, DECIDE ``!x. ~(x < x)``, num_CASES,
DIVIDES_RMUL,DIVIDES_LMUL,DIVIDES_REFL,ADD_CLAUSES]
QED | Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n) | Proof
rw [LESS_EQ_EXISTS]
>> Induct_on `p`
>> metis_tac [FACT, DECIDE ``!x. ~(x < x)``, num_CASES,
DIVIDES_RMUL,DIVIDES_LMUL,DIVIDES_REFL,ADD_CLAUSES]
QED | Theorem | theorem | LE_DIVIDES_FACT | examples | examples/euclid.sml | [] | [
"ADD_CLAUSES",
"DIVIDES_LMUL",
"DIVIDES_REFL",
"DIVIDES_RMUL",
"FACT",
"LESS_EQ_EXISTS",
"num_CASES",
"rw",
"x"
] | null | 151 | 158 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n)
Proof
Induct_on `n - m` >> rw []
>- metis_tac [EQ_LESS_EQ,DIVIDES_FACT]
>- (`?k. n = SUC k` by (Cases_on `n` >> fs[])
>> rw [FACT, DIVIDES_RMUL])
QED | Theorem LE_DIVIDES_FACT :
!m n. 0 < m /\ m <= n ==> m divides (FACT n) | Proof
Induct_on `n - m` >> rw []
>- metis_tac [EQ_LESS_EQ,DIVIDES_FACT]
>- (`?k. n = SUC k` by (Cases_on `n` >> fs[])
>> rw [FACT, DIVIDES_RMUL])
QED | Theorem | theorem | LE_DIVIDES_FACT | examples | examples/euclid.sml | [] | [
"DIVIDES_FACT",
"DIVIDES_RMUL",
"EQ_LESS_EQ",
"FACT",
"rw"
] | null | 160 | 167 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem NOT_PRIME_0 :
~prime 0
Proof
rw [prime_def,DIVIDES_ZERO]
QED | Theorem NOT_PRIME_0 :
~prime 0 | Proof
rw [prime_def,DIVIDES_ZERO]
QED | Theorem | theorem | NOT_PRIME_0 | examples | examples/euclid.sml | [] | [
"DIVIDES_ZERO",
"prime_def",
"rw"
] | null | 173 | 177 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem NOT_PRIME_1 :
~prime 1
Proof
rw [prime_def]
QED | Theorem NOT_PRIME_1 :
~prime 1 | Proof
rw [prime_def]
QED | Theorem | theorem | NOT_PRIME_1 | examples | examples/euclid.sml | [] | [
"prime_def",
"rw"
] | null | 179 | 183 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem PRIME_2 :
prime 2
Proof
rw [prime_def] >> drule DIVIDES_LE >> rw[]
QED | Theorem PRIME_2 :
prime 2 | Proof
rw [prime_def] >> drule DIVIDES_LE >> rw[]
QED | Theorem | theorem | PRIME_2 | examples | examples/euclid.sml | [] | [
"DIVIDES_LE",
"prime_def",
"rw"
] | null | 185 | 189 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem PRIME_POS :
!p. prime p ==> 0 < p
Proof
Cases >> rw [NOT_PRIME_0]
QED | Theorem PRIME_POS :
!p. prime p ==> 0 < p | Proof
Cases >> rw [NOT_PRIME_0]
QED | Theorem | theorem | PRIME_POS | examples | examples/euclid.sml | [] | [
"NOT_PRIME_0",
"rw"
] | null | 191 | 195 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem PRIME_FACTOR :
!n. ~(n = 1) ==> ?p. prime p /\ p divides n
Proof
completeInduct_on `n` >> rw [] >>
Cases_on `prime n`
>- metis_tac [DIVIDES_REFL]
>- (`?x. x divides n /\ x<>1 /\ x<>n` by metis_tac[prime_def]
>> metis_tac [LESS_OR_EQ, PRIME_2,
DIVIDES_LE, DIVIDES_TRANS, DI... | Theorem PRIME_FACTOR :
!n. ~(n = 1) ==> ?p. prime p /\ p divides n | Proof
completeInduct_on `n` >> rw [] >>
Cases_on `prime n`
>- metis_tac [DIVIDES_REFL]
>- (`?x. x divides n /\ x<>1 /\ x<>n` by metis_tac[prime_def]
>> metis_tac [LESS_OR_EQ, PRIME_2,
DIVIDES_LE, DIVIDES_TRANS, DIVIDES_ZERO])
QED | Theorem | theorem | PRIME_FACTOR | examples | examples/euclid.sml | [] | [
"DIVIDES_LE",
"DIVIDES_REFL",
"DIVIDES_TRANS",
"DIVIDES_ZERO",
"PRIME_2",
"prime_def",
"rw",
"x"
] | null | 208 | 217 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem PRIME_FACTOR :
!n. n<>1 ==> ?p. prime p /\ p divides n
Proof
completeInduct_on `n` >>
metis_tac [prime_def,LESS_OR_EQ, PRIME_2,
DIVIDES_REFL,DIVIDES_LE, DIVIDES_TRANS, DIVIDES_ZERO]
QED | Theorem PRIME_FACTOR :
!n. n<>1 ==> ?p. prime p /\ p divides n | Proof
completeInduct_on `n` >>
metis_tac [prime_def,LESS_OR_EQ, PRIME_2,
DIVIDES_REFL,DIVIDES_LE, DIVIDES_TRANS, DIVIDES_ZERO]
QED | Theorem | theorem | PRIME_FACTOR | examples | examples/euclid.sml | [] | [
"DIVIDES_LE",
"DIVIDES_REFL",
"DIVIDES_TRANS",
"DIVIDES_ZERO",
"PRIME_2",
"prime_def"
] | null | 224 | 230 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem EUCLID :
!n. ?p. n < p /\ prime p
Proof
spose_not_then strip_assume_tac
>> mp_tac (SPEC ``FACT n + 1`` PRIME_FACTOR)
>> rw [FACT_LESS, DECIDE ``x <> 0 <=> 0 < x``]
>> metis_tac [LE_DIVIDES_FACT, DIVIDES_ADDL, DIVIDES_ONE,
NOT_PRIME_1, NOT_LESS, PRIME_POS]
QED | Theorem EUCLID :
!n. ?p. n < p /\ prime p | Proof
spose_not_then strip_assume_tac
>> mp_tac (SPEC ``FACT n + 1`` PRIME_FACTOR)
>> rw [FACT_LESS, DECIDE ``x <> 0 <=> 0 < x``]
>> metis_tac [LE_DIVIDES_FACT, DIVIDES_ADDL, DIVIDES_ONE,
NOT_PRIME_1, NOT_LESS, PRIME_POS]
QED | Theorem | theorem | EUCLID | examples | examples/euclid.sml | [] | [
"DIVIDES_ADDL",
"DIVIDES_ONE",
"FACT",
"FACT_LESS",
"LE_DIVIDES_FACT",
"NOT_LESS",
"NOT_PRIME_1",
"PRIME_FACTOR",
"PRIME_POS",
"rw",
"x"
] | null | 241 | 249 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
Theorem EUCLID_AGAIN[local]:
!n. ?p. n < p /\ prime p
Proof
CCONTR_TAC >>
`?n. !p. n < p ==> ~prime p`
by metis_tac[] >>
`FACT n + 1 ≠ 1` by rw [FACT_LESS, DECIDE ``x<>0 <=> 0<x``] >>
‘∃p. prime p ∧ p divides (FACT n + 1)’
by metis_tac [PRIME_FACTOR] >>
... | Theorem EUCLID_AGAIN[local]:
!n. ?p. n < p /\ prime p | Proof
CCONTR_TAC >>
`?n. !p. n < p ==> ~prime p`
by metis_tac[] >>
`FACT n + 1 ≠ 1` by rw [FACT_LESS, DECIDE ``x<>0 <=> 0<x``] >>
‘∃p. prime p ∧ p divides (FACT n + 1)’
by metis_tac [PRIME_FACTOR] >>
`0 < p` by metis_tac [PRIME_POS] >>
... | Theorem | theorem | EUCLID_AGAIN | examples | examples/euclid.sml | [] | [
"DIVIDES_ADDL",
"DIVIDES_ONE",
"FACT",
"FACT_LESS",
"LE_DIVIDES_FACT",
"NOT_LESS",
"NOT_PRIME_1",
"PRIME_FACTOR",
"PRIME_POS",
"rw",
"x"
] | null | 258 | 273 | true | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
(isScalar(Scalar n) = T) /\ (isScalar(Array a) = F) | (isScalar(Scalar n) = T) /\ (isScalar(Array a) = F) | Define | definition | isScalar_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"T"
] | null | 30 | 30 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
ScalarOf(Scalar n) = n | ScalarOf(Scalar n) = n | Define | definition | ScalarOf_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 36 | 36 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(isArray(Array a) = T) /\ (isArray(Scalar n) = F) | (isArray(Array a) = T) /\ (isArray(Scalar n) = F) | Define | definition | isArray_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"T"
] | null | 42 | 42 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
ArrayOf(Array a) = a | ArrayOf(Array a) = a | Define | definition | ArrayOf_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 48 | 48 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(neval (Var v) s = ScalarOf(s ' v)) /\
(neval (Arr a e) s = (ArrayOf(s ' a) ' (Num(neval e s)))) /\
(neval (Const c) s = c) /\
(neval (Plus e1 e2) s = integer$int_add (neval e1 s) (neval e2 s)) /\
(neval (Sub e1 e2) s = integer$int_sub (neval e1 s) (neval e2 s)) /\
(neval (Times e1 e2) s = integer$int_mu... | (neval (Var v) s = ScalarOf(s ' v)) /\
(neval (Arr a e) s = (ArrayOf(s ' a) ' (Num(neval e s)))) /\
(neval (Const c) s = c) /\
(neval (Plus e1 e2) s = integer$int_add (neval e1 s) (neval e2 s)) /\
(neval (Sub e1 e2) s = integer$int_sub (neval e1 s) (neval e2 s)) /\
(neval (Times e1 e2) s = integer$int_mu... | Define | definition | neval_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"Num"
] | null | 99 | 99 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(safe_neval (Var v) s = if v IN FDOM s /\ isScalar (s ' v) then ScalarOf(s ' v) else 0i) /\
(safe_neval (Arr a e) s = if a IN FDOM s /\ isArray (s ' a) /\ integer$int_le 0i (safe_neval e s) /\ Num (safe_neval e s) IN FDOM (ArrayOf (s ' a)) then (ArrayOf(s ' a) ' (Num(safe_neval e s))) else 0i) /\
(safe_neval (Con... | (safe_neval (Var v) s = if v IN FDOM s /\ isScalar (s ' v) then ScalarOf(s ' v) else 0i) /\
(safe_neval (Arr a e) s = if a IN FDOM s /\ isArray (s ' a) /\ integer$int_le 0i (safe_neval e s) /\ Num (safe_neval e s) IN FDOM (ArrayOf (s ' a)) then (ArrayOf(s ' a) ' (Num(safe_neval e s))) else 0i) /\
(safe_neval (Con... | Define | definition | safe_neval_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"IN",
"Num",
"i",
"int_le"
] | null | 110 | 110 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(nevaluates (Var v) s = v IN FDOM s /\ isScalar (s ' v)) /\
(nevaluates (Arr a e) s = a IN FDOM s /\ isArray (s ' a) /\ integer$int_le 0i (safe_neval e s) /\ nevaluates e s /\ Num (safe_neval e s) IN FDOM (ArrayOf (s ' a))) /\
(nevaluates (Const c) s = T) /\
(nevaluates (Plus e1 e2) s = ne... | (nevaluates (Var v) s = v IN FDOM s /\ isScalar (s ' v)) /\
(nevaluates (Arr a e) s = a IN FDOM s /\ isArray (s ' a) /\ integer$int_le 0i (safe_neval e s) /\ nevaluates e s /\ Num (safe_neval e s) IN FDOM (ArrayOf (s ' a))) /\
(nevaluates (Const c) s = T) /\
(nevaluates (Plus e1 e2) s = ne... | Define | definition | nevaluates_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"IN",
"Num",
"T",
"i",
"int_le"
] | null | 131 | 131 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!e s. nevaluates e s ==> (safe_neval e s = neval e s) | `!e s. nevaluates e s ==> (safe_neval e s = neval e s) | store_thm | other | neval_safe_theorem1 | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 145 | 145 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`ONE_ONE str | `ONE_ONE str | prove | theorem | ONE_ONE_str | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"ONE_ONE"
] | null | 153 | 153 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`ONE_ONE nat | `ONE_ONE nat | prove | theorem | ONE_ONE_nat | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"ONE_ONE"
] | null | 159 | 159 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(beval (Equal e1 e2) s = (neval e1 s = neval e2 s)) /\
(beval (Less e1 e2) s = integer$int_lt (neval e1 s) (neval e2 s)) /\
(beval (LessEq e1 e2) s = integer$int_le (neval e1 s) (neval e2 s)) /\
(beval (Greater e1 e2) s = integer$int_gt (neval e1 s) (neval e2 s)) /\
(beval (GreaterEq e1 e2) s = integer$int_... | (beval (Equal e1 e2) s = (neval e1 s = neval e2 s)) /\
(beval (Less e1 e2) s = integer$int_lt (neval e1 s) (neval e2 s)) /\
(beval (LessEq e1 e2) s = integer$int_le (neval e1 s) (neval e2 s)) /\
(beval (Greater e1 e2) s = integer$int_gt (neval e1 s) (neval e2 s)) /\
(beval (GreaterEq e1 e2) s = integer$int_... | Define | definition | beval_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"int_ge",
"int_gt",
"int_le"
] | null | 164 | 164 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(safe_beval (Equal e1 e2) s = (safe_neval e1 s = safe_neval e2 s)) /\
(safe_beval (Less e1 e2) s = integer$int_lt (safe_neval e1 s) (safe_neval e2 s)) /\
(safe_beval (LessEq e1 e2) s = integer$int_le (safe_neval e1 s) (safe_neval e2 s)) /\
(safe_beval (Greater e1 e2) s = integer$int_gt (safe_neval e1 s) (safe_... | (safe_beval (Equal e1 e2) s = (safe_neval e1 s = safe_neval e2 s)) /\
(safe_beval (Less e1 e2) s = integer$int_lt (safe_neval e1 s) (safe_neval e2 s)) /\
(safe_beval (LessEq e1 e2) s = integer$int_le (safe_neval e1 s) (safe_neval e2 s)) /\
(safe_beval (Greater e1 e2) s = integer$int_gt (safe_neval e1 s) (safe_... | Define | definition | safe_beval_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"int_ge",
"int_gt",
"int_le"
] | null | 179 | 179 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(bevaluates (Equal e1 e2) s = nevaluates e1 s /\ nevaluates e2 s) /\
(bevaluates (Less e1 e2) s = nevaluates e1 s /\ nevaluates e2 s) /\
(bevaluates (LessEq e1 e2) s = nevaluates e1 s /\ nevaluates e2 s) /\
(bevaluates (Greater e1 e2) s = nevaluates e1 s /\ nevaluates e2 s) /\
(bevaluates (GreaterEq e1 e2) ... | (bevaluates (Equal e1 e2) s = nevaluates e1 s /\ nevaluates e2 s) /\
(bevaluates (Less e1 e2) s = nevaluates e1 s /\ nevaluates e2 s) /\
(bevaluates (LessEq e1 e2) s = nevaluates e1 s /\ nevaluates e2 s) /\
(bevaluates (Greater e1 e2) s = nevaluates e1 s /\ nevaluates e2 s) /\
(bevaluates (GreaterEq e1 e2) ... | Define | definition | bevaluates_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 190 | 190 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!e s. bevaluates e s ==> (safe_beval e s = beval e s) | `!e s. bevaluates e s ==> (safe_beval e s = beval e s) | store_thm | other | beval_safe_theorem1 | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 201 | 201 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`ONE_ONE str | `ONE_ONE str | prove | theorem | ONE_ONE_str | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"ONE_ONE"
] | null | 205 | 205 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`ONE_ONE nat | `ONE_ONE nat | prove | theorem | ONE_ONE_nat | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"ONE_ONE"
] | null | 207 | 207 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(aeval (ArrConst f) s = f) /\
(aeval (ArrVar v) s = ArrayOf(s ' v)) /\
(aeval (ArrUpdate a e1 e2) s = aeval a s |+ (Num(neval e1 s), neval e2 s)) | (aeval (ArrConst f) s = f) /\
(aeval (ArrVar v) s = ArrayOf(s ' v)) /\
(aeval (ArrUpdate a e1 e2) s = aeval a s |+ (Num(neval e1 s), neval e2 s)) | Define | definition | aeval_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"Num"
] | null | 212 | 212 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(aevaluates (ArrConst f) s = T) /\
(aevaluates (ArrVar v) s = v IN FDOM s) /\
(aevaluates (ArrUpdate a e1 e2) s = aevaluates a s /\ nevaluates e1 s /\ integer$int_le 0i (safe_neval e1 s) /\ nevaluates e2 s) | (aevaluates (ArrConst f) s = T) /\
(aevaluates (ArrVar v) s = v IN FDOM s) /\
(aevaluates (ArrUpdate a e1 e2) s = aevaluates a s /\ nevaluates e1 s /\ integer$int_le 0i (safe_neval e1 s) /\ nevaluates e2 s) | Define | definition | aevaluates_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"IN",
"T",
"i",
"int_le"
] | null | 223 | 223 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(safe_aeval (ArrConst f) s = f) /\
(safe_aeval (ArrVar v) s = if v IN FDOM s then ArrayOf(s ' v) else FEMPTY) /\
(safe_aeval (ArrUpdate a e1 e2) s = if integer$int_le 0i (safe_neval e1 s) then safe_aeval a s |+ (Num(safe_neval e1 s), safe_neval e2 s) else safe_aeval a s) | (safe_aeval (ArrConst f) s = f) /\
(safe_aeval (ArrVar v) s = if v IN FDOM s then ArrayOf(s ' v) else FEMPTY) /\
(safe_aeval (ArrUpdate a e1 e2) s = if integer$int_le 0i (safe_neval e1 s) then safe_aeval a s |+ (Num(safe_neval e1 s), safe_neval e2 s) else safe_aeval a s) | Define | definition | safe_aeval_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"IN",
"Num",
"i",
"int_le"
] | null | 233 | 233 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!e s. aevaluates e s ==> (safe_aeval e s = aeval e s) | `!e s. aevaluates e s ==> (safe_aeval e s = aeval e s) | store_thm | other | aeval_safe_theorem1 | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 243 | 243 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`ONE_ONE str | `ONE_ONE str | prove | theorem | ONE_ONE_str | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"ONE_ONE"
] | null | 251 | 251 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`ONE_ONE nat | `ONE_ONE nat | prove | theorem | ONE_ONE_nat | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"ONE_ONE"
] | null | 257 | 257 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(naeval (INL e) s = Scalar(neval e s)) /\
(naeval (INR a) s = Array(aeval a s)) | (naeval (INL e) s = Scalar(neval e s)) /\
(naeval (INR a) s = Array(aeval a s)) | Define | definition | naeval_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"INL",
"INR"
] | null | 261 | 261 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(safe_naeval (INL e) s = Scalar(safe_neval e s)) /\
(safe_naeval (INR a) s = Array(safe_aeval a s)) | (safe_naeval (INL e) s = Scalar(safe_neval e s)) /\
(safe_naeval (INR a) s = Array(safe_aeval a s)) | Define | definition | safe_naeval_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"INL",
"INR"
] | null | 270 | 270 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(naevaluates (INL e) s = nevaluates e s) /\
(naevaluates (INR a) s = aevaluates a s) | (naevaluates (INL e) s = nevaluates e s) /\
(naevaluates (INR a) s = aevaluates a s) | Define | definition | naevaluates_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"INL",
"INR"
] | null | 275 | 275 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!e s. naevaluates e s ==> (safe_naeval e s = naeval e s) | `!e s. naevaluates e s ==> (safe_naeval e s = naeval e s) | store_thm | other | naeval_safe_theorem1 | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 280 | 280 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
Update v e s = s |+ (v, naeval e s) | Update v e s = s |+ (v, naeval e s) | Define | definition | Update_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"Update"
] | null | 286 | 286 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
safe_Update v e s = s |+ (v, safe_naeval e s) | safe_Update v e s = s |+ (v, safe_naeval e s) | Define | definition | safe_Update_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 294 | 294 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
Updates v e s = naevaluates e s | Updates v e s = naevaluates e s | Define | definition | Updates_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 298 | 298 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`Updates v e s ==> (safe_Update v e s = Update v e s) | `Updates v e s ==> (safe_Update v e s = Update v e s) | store_thm | other | Updates_safe_theorem1 | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"Update"
] | null | 302 | 302 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`(Update v (INL e) s = s |+ (v, Scalar(neval e s))) /\
(Update v (INR a) s = s |+ (v, Array(aeval a s))) | `(Update v (INL e) s = s |+ (v, Scalar(neval e s))) /\
(Update v (INR a) s = s |+ (v, Array(aeval a s))) | store_thm | other | UpdateCases | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"INL",
"INR",
"Update"
] | null | 308 | 308 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(Exp(Scalar n) = INL(Const n)) /\
(Exp(Array f) = INR(ArrConst f)) | (Exp(Scalar n) = INL(Const n)) /\
(Exp(Array f) = INR(ArrConst f)) | Define | definition | Exp_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"INL",
"INR"
] | null | 315 | 315 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!v val s. Update v (Exp val) s = s |+ (v, val) | `!v val s. Update v (Exp val) s = s |+ (v, val) | store_thm | other | Update_Exp | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"Update"
] | null | 321 | 321 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
While a c = AnWhile a ARB c | While a c = AnWhile a ARB c | Define | definition | While_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 348 | 348 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
Assign v e = GenAssign v (INL e) | Assign v e = GenAssign v (INL e) | Define | definition | Assign_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"INL"
] | null | 352 | 352 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
ArrayAssign v e1 e2 = GenAssign v (INR(ArrUpdate (ArrVar v) e1 e2)) | ArrayAssign v e1 e2 = GenAssign v (INR(ArrUpdate (ArrVar v) e1 e2)) | Define | definition | ArrayAssign_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"INR"
] | null | 360 | 360 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
(Locals [] c = c) /\
(Locals (v::vl) c = Local v (Locals vl c)) | (Locals [] c = c) /\
(Locals (v::vl) c = Local v (Locals vl c)) | Define | definition | Locals_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 368 | 368 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s2. EVAL Skip s1 s2 = (s1 = s2) | `!s1 s2. EVAL Skip s1 s2 = (s1 = s2) | store_thm | other | SKIP_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 457 | 457 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s. EVAL Skip s s | `!s. EVAL Skip s s | store_thm | other | SKIP | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 470 | 470 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s2 v e. EVAL (GenAssign v e) s1 s2 = (s2 = Update v e s1) | `!s1 s2 v e. EVAL (GenAssign v e) s1 s2 = (s2 = Update v e s1) | store_thm | other | GEN_ASSIGN_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"Update"
] | null | 475 | 475 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s v e. EVAL (GenAssign v e) s (Update v e s) | `!s v e. EVAL (GenAssign v e) s (Update v e s) | store_thm | other | GEN_ASSIGN | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"Update"
] | null | 480 | 480 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s v e. EVAL (Assign v e) s (Update v (INL e) s) | `!s v e. EVAL (Assign v e) s (Update v (INL e) s) | store_thm | other | ASSIGN | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"INL",
"Update"
] | null | 485 | 485 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s v e1 e2.
EVAL (ArrayAssign v e1 e2) s (Update v (INR(ArrUpdate (ArrVar v) e1 e2)) s) | `!s v e1 e2.
EVAL (ArrayAssign v e1 e2) s (Update v (INR(ArrUpdate (ArrVar v) e1 e2)) s) | store_thm | other | ARRAY_ASSIGN | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"INR",
"Update"
] | null | 490 | 490 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s2 v. EVAL (Dispose v) s1 s2 = (s2 = s1 \\ v) | `!s1 s2 v. EVAL (Dispose v) s1 s2 = (s2 = s1 \\ v) | store_thm | other | DISPOSE_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 496 | 496 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s v. EVAL (Dispose v) s (s \\ v) | `!s v. EVAL (Dispose v) s (s \\ v) | store_thm | other | DISPOSE | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 501 | 501 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s3 c1 c2. EVAL (Seq c1 c2) s1 s3 = ?s2. EVAL c1 s1 s2 /\ EVAL c2 s2 s3 | `!s1 s3 c1 c2. EVAL (Seq c1 c2) s1 s3 = ?s2. EVAL c1 s1 s2 /\ EVAL c2 s2 s3 | store_thm | other | SEQ_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 506 | 506 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s2 b c1 c2.
beval b s1 ==> (EVAL (Cond b c1 c2) s1 s2 = EVAL c1 s1 s2) | `!s1 s2 b c1 c2.
beval b s1 ==> (EVAL (Cond b c1 c2) s1 s2 = EVAL c1 s1 s2) | store_thm | other | IF_T_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 511 | 511 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s2 b c1 c2.
~beval b s1 ==> (EVAL (Cond b c1 c2) s1 s2 = EVAL c2 s1 s2) | `!s1 s2 b c1 c2.
~beval b s1 ==> (EVAL (Cond b c1 c2) s1 s2 = EVAL c2 s1 s2) | store_thm | other | IF_F_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 517 | 517 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s2 b s1 s2.
EVAL (Cond b c1 c2) s1 s2 =
if beval b s1 then EVAL c1 s1 s2 else EVAL c2 s1 s2 | `!s1 s2 b s1 s2.
EVAL (Cond b c1 c2) s1 s2 =
if beval b s1 then EVAL c1 s1 s2 else EVAL c2 s1 s2 | store_thm | other | IF_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 523 | 523 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s3 b n c.
beval b s1 ==>
(EVAL (AnWhile b n c) s1 s3 =
?s2. EVAL c s1 s2 /\ EVAL (AnWhile b n c) s2 s3) | `!s1 s3 b n c.
beval b s1 ==>
(EVAL (AnWhile b n c) s1 s3 =
?s2. EVAL c s1 s2 /\ EVAL (AnWhile b n c) s2 s3) | store_thm | other | ANWHILE_T_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 531 | 531 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s3 b c.
beval b s1 ==>
(EVAL (While b c) s1 s3 =
?s2. EVAL c s1 s2 /\ EVAL (While b c) s2 s3) | `!s1 s3 b c.
beval b s1 ==>
(EVAL (While b c) s1 s3 =
?s2. EVAL c s1 s2 /\ EVAL (While b c) s2 s3) | store_thm | other | WHILE_T_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 540 | 540 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s2 b n c. ~beval b s1 ==> (EVAL (AnWhile b n c) s1 s2 = (s1 = s2)) | `!s1 s2 b n c. ~beval b s1 ==> (EVAL (AnWhile b n c) s1 s2 = (s1 = s2)) | store_thm | other | ANWHILE_F_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 549 | 549 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s2 b c. ~beval b s1 ==> (EVAL (While b c) s1 s2 = (s1 = s2)) | `!s1 s2 b c. ~beval b s1 ==> (EVAL (While b c) s1 s2 = (s1 = s2)) | store_thm | other | WHILE_F_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 555 | 555 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s3 b n c.
EVAL (AnWhile b n c) s1 s3 =
if beval b s1
then ?s2. EVAL c s1 s2 /\ EVAL (AnWhile b n c) s2 s3
else (s1 = s3) | `!s1 s3 b n c.
EVAL (AnWhile b n c) s1 s3 =
if beval b s1
then ?s2. EVAL c s1 s2 /\ EVAL (AnWhile b n c) s2 s3
else (s1 = s3) | store_thm | other | ANWHILE_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 561 | 561 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s3 b c.
EVAL (While b c) s1 s3 =
if beval b s1
then ?s2. EVAL c s1 s2 /\ EVAL (While b c) s2 s3
else (s1 = s3) | `!s1 s3 b c.
EVAL (While b c) s1 s3 =
if beval b s1
then ?s2. EVAL c s1 s2 /\ EVAL (While b c) s2 s3
else (s1 = s3) | store_thm | other | WHILE_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 570 | 570 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!s1 s2 v c.
EVAL (Local v c) s1 s2 =
?s3. EVAL c s1 s3
/\
(s2 = if v IN FDOM s1 then s3 |+ (v, (s1 ' v)) else s3 \\ v) | `!s1 s2 v c.
EVAL (Local v c) s1 s2 =
?s3. EVAL c s1 s3
/\
(s2 = if v IN FDOM s1 then s3 |+ (v, (s1 ' v)) else s3 \\ v) | store_thm | other | LOCAL_THM | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"IN"
] | null | 579 | 579 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!c1 c2 c3 s1 s2.
EVAL (Seq (Seq c1 c2) c3) s1 s2 = EVAL (Seq c1 (Seq c2 c3)) s1 s2 | `!c1 c2 c3 s1 s2.
EVAL (Seq (Seq c1 c2) c3) s1 s2 = EVAL (Seq c1 (Seq c2 c3)) s1 s2 | store_thm | other | SEQ_ASSOC | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 588 | 588 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!c st1 st2. EVAL c st1 st2 ==> !st3. EVAL c st1 st3 ==> (st2 = st3) | `!c st1 st2. EVAL c st1 st2 ==> !st3. EVAL c st1 st3 ==> (st2 = st3) | store_thm | other | EVAL_DETERMINISTIC | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"st2",
"st3"
] | null | 597 | 597 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!c st1 st2 p.
(p st1 ==> EVAL c st1 st2) ==> !st3. EVAL c st1 st3 ==> p st1 ==> (st2 = st3) | `!c st1 st2 p.
(p st1 ==> EVAL c st1 st2) ==> !st3. EVAL c st1 st3 ==> p st1 ==> (st2 = st3) | store_thm | other | IMP_EVAL_DETERMINISTIC | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"st2",
"st3"
] | null | 608 | 608 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
SPEC P c Q = !s1 s2. P s1 /\ EVAL c s1 s2 ==> Q s2 | SPEC P c Q = !s1 s2. P s1 /\ EVAL c s1 s2 ==> Q s2 | Define | definition | SPEC_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 616 | 616 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
RSPEC P c R = !s1 s2. P s1 /\ EVAL c s1 s2 ==> R s1 s2 | RSPEC P c R = !s1 s2. P s1 /\ EVAL c s1 s2 ==> R s1 s2 | Define | definition | RSPEC_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 624 | 624 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!A c f.
(!s. A s ==> EVAL c s (f s))
==>
!P R.
(!s. (P s ==> A s) /\ (A s ==> R s (f s))) ==> RSPEC P c R | `!A c f.
(!s. A s ==> EVAL c s (f s))
==>
!P R.
(!s. (P s ==> A s) /\ (A s ==> R s (f s))) ==> RSPEC P c R | store_thm | other | EVAL_RSPEC | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"A"
] | null | 633 | 633 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
IMP pre post = \prog. RSPEC pre prog post | IMP pre post = \prog. RSPEC pre prog post | Define | definition | IMP_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"pre"
] | null | 644 | 644 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
AND spec1 spec2 = \prog. spec1 prog /\ spec2 prog | AND spec1 spec2 = \prog. spec1 prog /\ spec2 prog | Define | definition | AND_def | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"AND"
] | null | 652 | 652 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!P. SPEC P Skip P | `!P. SPEC P Skip P | store_thm | other | SKIP_RULE | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 656 | 656 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!P v e.
SPEC (\s. P (s \\ v)) (Dispose v) P | `!P v e.
SPEC (\s. P (s \\ v)) (Dispose v) P | store_thm | other | DISPOSE_RULE | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 665 | 665 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!P v e.
SPEC (P o Update v e) (GenAssign v e) P | `!P v e.
SPEC (P o Update v e) (GenAssign v e) P | store_thm | other | GEN_ASSIGN_RULE | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [
"Update"
] | null | 675 | 675 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!P v.
SPEC (\s. P (s \\ v)) (Dispose v) P | `!P v.
SPEC (\s. P (s \\ v)) (Dispose v) P | store_thm | other | DISPOSE_RULE | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 685 | 685 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!P c1 c2 Q R.
SPEC P c1 Q /\ SPEC Q c2 R ==> SPEC P (Seq c1 c2) R | `!P c1 c2 Q R.
SPEC P c1 Q /\ SPEC Q c2 R ==> SPEC P (Seq c1 c2) R | store_thm | other | SEQ_RULE | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 695 | 695 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
`!P b c1 c2 Q.
SPEC (\s. P(s) /\ beval b s) c1 Q /\
SPEC (\s. P(s) /\ ~beval b s) c2 Q ==> SPEC P (Cond b c1 c2) Q | `!P b c1 c2 Q.
SPEC (\s. P(s) /\ beval b s) c1 Q /\
SPEC (\s. P(s) /\ ~beval b s) c2 Q ==> SPEC P (Cond b c1 c2) Q | store_thm | other | COND_RULE | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 705 | 705 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
!c s1 s2. EVAL c s1 s2 ==> !b' n' c'. (c = AnWhile b' n' c') ==> ~beval b' s2 | !c s1 s2. EVAL c s1 s2 ==> !b' n' c'. (c = AnWhile b' n' c') ==> ~beval b' s2 | prove | theorem | lemma1 | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 722 | 722 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof | |
!c s1 s2.
EVAL c s1 s2 ==>
!b' n' c'.
(c = AnWhile b' n' c') ==>
(!s1 s2. P s1 /\ beval b' s1 /\ EVAL c' s1 s2 ==> P s2)
==> (P s1 ==> P s2) | !c s1 s2.
EVAL c s1 s2 ==>
!b' n' c'.
(c = AnWhile b' n' c') ==>
(!s1 s2. P s1 /\ beval b' s1 /\ EVAL c' s1 s2 ==> P s2)
==> (P s1 ==> P s2) | prove | theorem | lemma2 | examples.acl2.examples | examples/acl2/examples/fmapExample.sml | [] | [] | null | 726 | 726 | false | https://github.com/HOL-Theorem-Prover/HOL | ab6450c497d986a09043c9e930c36841ba93fbdb | statement+proof |
End of preview. Expand in Data Studio
HOL4
A structured dataset of theorems and definitions from HOL4, a mature theorem prover for higher-order logic.
Source
- Repository: https://github.com/HOL-Theorem-Prover/HOL
- Commit:
ab6450c497d986a09043c9e930c36841ba93fbdb - Files: 2940
- License: bsd-3-clause
Schema
| Column | Type | Description |
|---|---|---|
| fact | string | Verbatim declaration: statement followed by proof where present |
| statement | string | Verbatim statement (keyword through the closing period) |
| proof | string | Verbatim proof block (Proof. ... Qed./Defined.), empty if none |
| type | string | Raw declaration keyword |
| kind | string | Normalized kind |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, null if absent |
| line_start | int | First source line |
| line_end | int | Last source line |
| has_proof | bool | Whether a proof block was captured |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
| content_level | string | statement+proof |
Statistics
- Entries: 75,265
- With proof: 54,981 (73.0%)
- With docstring: 0 (0.0%)
- Libraries: 350
By type
| Type | Count |
|---|---|
| Theorem | 55,351 |
| Definition | 12,661 |
| prove | 4,896 |
| Define | 756 |
| TAC_PROOF | 524 |
| store_thm | 410 |
| new_definition | 385 |
| Inductive | 148 |
| tDefine | 76 |
| new_recursive_definition | 22 |
| xDefine | 17 |
| CoInductive | 12 |
| zDefine | 5 |
| Triviality | 2 |
Example
Theorem DIVIDES_ONE:
!x. x divides 1 <=> x = 1
Proof
metis_tac [divides_def,MULT_CLAUSES,MULT_EQ_1]
QED
- kind: theorem | symbolic_name:
DIVIDES_ONE| examples/euclid.sml:51
Use
Statement and proof are available both joined (fact) and split (statement, proof) for
proof-term modeling, autoformalization, retrieval, and dependency analysis via deps.
Citation
@misc{hol4_dataset,
title = {HOL4},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/HOL-Theorem-Prover/HOL, commit ab6450c497d9},
url = {https://huggingface.co/datasets/phanerozoic/HOL4}
}
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