Thm* a:T, Q:(T![](FONT/dash.png) Prop). (a = !x:T. Q(x)) ![](FONT/eq.png) a {!x:T | Q(x)} | [uni_sat_imp_in_uni_set] |
Thm* (P ![](FONT/eq.png) Q) ![](FONT/eq.png) (Q ![](FONT/eq.png) P) ![](FONT/eq.png) (P ![](FONT/if_big.png) Q) | [implies_antisymmetry] |
Thm* (P ![](FONT/if_big.png) Q) ![](FONT/eq.png) ( P ![](FONT/if_big.png) Q) | [squash_functionality_wrt_iff] |
Thm* (P ![](FONT/eq.png) Q) ![](FONT/eq.png) P ![](FONT/eq.png) Q | [squash_functionality_wrt_implies] |
Thm* (P1 ![](FONT/eq.png) P2) ![](FONT/eq.png) (Q1 ![](FONT/eq.png) Q2) ![](FONT/eq.png) P1 Q1 ![](FONT/eq.png) P2 Q2 | [or_functionality_wrt_implies] |
Thm* (P1 ![](FONT/if_big.png) P2) ![](FONT/eq.png) (Q1 ![](FONT/if_big.png) Q2) ![](FONT/eq.png) (P1 Q1 ![](FONT/if_big.png) P2 Q2) | [or_functionality_wrt_iff] |
Thm* (P ![](FONT/if_big.png) Q) ![](FONT/eq.png) P ![](FONT/eq.png) Q | [not_functionality_wrt_implies] |
Thm* (P ![](FONT/if_big.png) Q) ![](FONT/eq.png) ( P ![](FONT/if_big.png) Q) | [not_functionality_wrt_iff] |
Thm* P,Q:(S![](FONT/dash.png) Prop).
Thm* S = T ![](FONT/eq.png) ( x:S. P(x) ![](FONT/eq.png) Q(x)) ![](FONT/eq.png) ( x:S. P(x)) ![](FONT/eq.png) ( y:T. Q(y)) | [exists_functionality_wrt_implies] |
Thm* P,Q:(S![](FONT/dash.png) Prop).
Thm* S = T ![](FONT/eq.png) ( x:S. P(x) ![](FONT/if_big.png) Q(x)) ![](FONT/eq.png) (( x:S. P(x)) ![](FONT/if_big.png) ( y:T. Q(y))) | [exists_functionality_wrt_iff] |
Thm* P,Q:(S![](FONT/dash.png) Prop).
Thm* S = T ![](FONT/eq.png) ( z:S. P(z) ![](FONT/eq.png) Q(z)) ![](FONT/eq.png) ( x:S. P(x)) ![](FONT/eq.png) ( y:T. Q(y)) | [all_functionality_wrt_implies] |
Thm* P,Q:(S![](FONT/dash.png) Prop).
Thm* S = T ![](FONT/eq.png) ( x:S. P(x) ![](FONT/if_big.png) Q(x)) ![](FONT/eq.png) (( x:S. P(x)) ![](FONT/if_big.png) ( y:T. Q(y))) | [all_functionality_wrt_iff] |
Thm* (P1 ![](FONT/if_big.png) P2) ![](FONT/eq.png) (Q1 ![](FONT/if_big.png) Q2) ![](FONT/eq.png) ((P1 ![](FONT/if_big.png) Q1) ![](FONT/if_big.png) (P2 ![](FONT/if_big.png) Q2)) | [iff_functionality_wrt_iff] |
Thm* (P ![](FONT/if_big.png) Q) ![](FONT/eq.png) (Dec(P) ![](FONT/if_big.png) Dec(Q)) | [decidable_functionality] |
Thm* (P1 ![](FONT/if_big.png) P2) ![](FONT/eq.png) (Q1 ![](FONT/eq.png) Q2) ![](FONT/eq.png) (P1 ![](FONT/eq.png) Q1) ![](FONT/eq.png) P2 ![](FONT/eq.png) Q2 | [implies_functionality_wrt_implies] |
Thm* (P1 ![](FONT/if_big.png) P2) ![](FONT/eq.png) (Q1 ![](FONT/if_big.png) Q2) ![](FONT/eq.png) ((P1 ![](FONT/eq.png) Q1) ![](FONT/if_big.png) (P2 ![](FONT/eq.png) Q2)) | [implies_functionality_wrt_iff] |
Thm* (P1 ![](FONT/eq.png) P2) ![](FONT/eq.png) (Q1 ![](FONT/eq.png) Q2) ![](FONT/eq.png) P1 & Q1 ![](FONT/eq.png) P2 & Q2 | [and_functionality_wrt_implies] |
Thm* (P1 ![](FONT/if_big.png) P2) ![](FONT/eq.png) (Q1 ![](FONT/if_big.png) Q2) ![](FONT/eq.png) (P1 & Q1 ![](FONT/if_big.png) P2 & Q2) | [and_functionality_wrt_iff] |
Thm* (P ![](FONT/eq.png) Q) ![](FONT/eq.png) (Q ![](FONT/eq.png) R) ![](FONT/eq.png) P ![](FONT/eq.png) R | [implies_transitivity] |
Thm* (P ![](FONT/if_big.png) Q) ![](FONT/eq.png) (Q ![](FONT/if_big.png) R) ![](FONT/eq.png) (P ![](FONT/if_big.png) R) | [iff_transitivity] |
Thm* Dec(A) ![](FONT/eq.png) ( (A & B) ![](FONT/if_big.png) A B) | [not_over_and] |
Thm* A B ![](FONT/eq.png) (A & B) | [not_over_and_b] |
Thm* Dec(A) ![](FONT/eq.png) (![](FONT/not.png) A ![](FONT/if_big.png) A) | [dneg_elim_a] |
Thm* Dec(A) ![](FONT/eq.png) ![](FONT/not.png) A ![](FONT/eq.png) A | [dneg_elim] |
Thm* SqStable(P) ![](FONT/eq.png) ( P ![](FONT/if_big.png) P) | [squash_elim] |
Thm* Dec(P) ![](FONT/eq.png) SqStable(P) | [sq_stable_from_decidable] |
Thm* Stable{P} ![](FONT/eq.png) SqStable(P) | [sq_stable__from_stable] |
Thm* P:(A![](FONT/dash.png) Prop). ( x:A. SqStable(P(x))) ![](FONT/eq.png) SqStable( x:A. P(x)) | [sq_stable__all] |
Thm* SqStable(P) ![](FONT/eq.png) SqStable(Q) ![](FONT/eq.png) SqStable(P ![](FONT/if_big.png) Q) | [sq_stable__iff] |
Thm* SqStable(Q) ![](FONT/eq.png) SqStable(P ![](FONT/eq.png) Q) | [sq_stable__implies] |
Thm* SqStable(P) ![](FONT/eq.png) SqStable(Q) ![](FONT/eq.png) SqStable(P & Q) | [sq_stable__and] |
Thm* Dec(P) ![](FONT/eq.png) Stable{P} | [stable__from_decidable] |
Thm* f,g:(A![](FONT/dash.png) B). ( x:A. Stable{f(x) = g(x)}) ![](FONT/eq.png) Stable{f = g} | [stable__function_equal] |
Thm* Dec(A) ![](FONT/eq.png) (A ![](FONT/if_big.png) B) ![](FONT/eq.png) Dec(B) | [iff_preserves_decidability] |
Thm* Dec(P) ![](FONT/eq.png) Dec(Q) ![](FONT/eq.png) Dec(P ![](FONT/if_big.png) Q) | [decidable__iff] |
Thm* Dec(P) ![](FONT/eq.png) Dec(Q) ![](FONT/eq.png) Dec(P ![](FONT/eq.png) Q) | [decidable__implies] |
Thm* Dec(P) ![](FONT/eq.png) Dec(Q) ![](FONT/eq.png) Dec(P & Q) | [decidable__and] |
Thm* Dec(P) ![](FONT/eq.png) Dec(Q) ![](FONT/eq.png) Dec(P Q) | [decidable__or] |
Def a = !x:T. Q(x) == Q(a) & ( a':T. Q(a') ![](FONT/eq.png) a' = a) | [uni_sat] |
Def {!x:T | P(x)} == {x:T| P(x) & ( y:T. P(y) ![](FONT/eq.png) y = x) } | [unique_set] |
Def SqStable(P) == P ![](FONT/eq.png) P | [sq_stable] |
Def Stable{P} == ![](FONT/not.png) P ![](FONT/eq.png) P | [stable] |
Def P ![](FONT/if_big.png) Q == (P ![](FONT/eq.png) Q) & (P ![](FONT/if_big.png) Q) | [iff] |
Def A == A ![](FONT/eq.png) False | [not] |
Def P ![](FONT/if_big.png) Q == Q ![](FONT/eq.png) P | [rev_implies] |