Nuprl Lemma : perm_induction
∀n:ℕ. ∀Q:Sym(n) ⟶ ℙ. (Q[id_perm()] ⇒ (∀p:Sym(n). (Q[p] ⇒ (∀i,j:ℕn. Q[txpose_perm(i;j) O p]))) ⇒ {∀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 :
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
sym_grp: Sym(n),
uall: ∀[x:A]. B[x],
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
perm_igrp: perm_igrp(T),
mk_igrp: mk_igrp(T;op;id;inv),
grp_car: |g|,
pi1: fst(t),
uimplies: b supposing a,
int_seg: {i..j-},
top: Top,
mon_reduce: mon_reduce,
grp_id: e,
pi2: snd(t),
infix_ap: x f y,
grp_op: *
Lemmas referenced :
perm_wf,
int_seg_wf,
subtype_rel_self,
comp_perm_wf,
txpose_perm_wf,
id_perm_wf,
nat_wf,
sym_grp_is_swaps,
list_induction,
all_wf,
equal_wf,
mon_reduce_wf,
perm_igrp_wf,
map_wf,
grp_car_wf,
list_wf,
list_subtype_base,
product_subtype_base,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
le_wf,
map_nil_lemma,
istype-void,
reduce_nil_lemma,
map_cons_lemma,
reduce_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
functionIsType,
applyEquality,
instantiate,
universeEquality,
because_Cache,
inhabitedIsType,
productElimination,
independent_functionElimination,
productEquality,
lambdaEquality_alt,
functionEquality,
productIsType,
equalityIsType3,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
intEquality,
isect_memberEquality_alt,
voidElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}Q:Sym(n) {}\mrightarrow{} \mBbbP{}.
(Q[id\_perm()] {}\mRightarrow{} (\mforall{}p:Sym(n). (Q[p] {}\mRightarrow{} (\mforall{}i,j:\mBbbN{}n. Q[txpose\_perm(i;j) O p]))) {}\mRightarrow{} \{\mforall{}p:Sym(n). Q[p]\})
Date html generated:
2019_10_16-PM-01_02_05
Last ObjectModification:
2018_10_08-AM-11_55_45
Theory : list_2
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