Nuprl Lemma : perm_induction_b
∀n:ℕ. ∀Q:Sym(n) ⟶ ℙ. (Q[id_perm()] ⇒ (∀p:Sym(n). (Q[p] ⇒ (∀i:ℕ+n. Q[p O txpose_perm(i;0)]))) ⇒ {∀p:Sym(n). Q[p]})
Proof
Definitions occuring in Statement :
txpose_perm: txpose_perm,
sym_grp: Sym(n),
comp_perm: comp_perm,
id_perm: id_perm(),
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
sym_grp: Sym(n),
uall: ∀[x:A]. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True
Lemmas referenced :
perm_wf,
int_seg_wf,
subtype_rel_self,
comp_perm_wf,
txpose_perm_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
istype-false,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
id_perm_wf,
nat_wf,
perm_induction_a,
inv_perm_wf,
perm_grp_inv_id,
iff_weakening_equal,
perm_grp_inv_thru_op,
squash_wf,
true_wf,
txpose_perm_inv,
perm_grp_inv_inv
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
universeEquality,
because_Cache,
dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
equalityTransitivity,
equalitySymmetry,
imageElimination,
inhabitedIsType,
imageMemberEquality,
baseClosed
Latex:
\mforall{}n:\mBbbN{}. \mforall{}Q:Sym(n) {}\mrightarrow{} \mBbbP{}.
(Q[id\_perm()] {}\mRightarrow{} (\mforall{}p:Sym(n). (Q[p] {}\mRightarrow{} (\mforall{}i:\mBbbN{}\msupplus{}n. Q[p O txpose\_perm(i;0)]))) {}\mRightarrow{} \{\mforall{}p:Sym(n). Q[p]\})
Date html generated:
2019_10_16-PM-01_02_08
Last ObjectModification:
2018_10_08-PM-05_44_42
Theory : list_2
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