hcju/isabelleps huggingface dataset:
simps_4: fixes \<phi> :: "('d, 'e, 'f) formula"   and \<psi> :: "('d, 'e, 'f) formula"   and \<sigma> :: "('d, 'e, 'f) subst" shows "Fsubst (\<phi> && \<psi>) \<sigma> = (Fsubst \<phi> \<sigma> && Fsubst \<psi> \<sigma>)"
simps_2: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and u :: "'f"   and x :: "'e"   and xs :: "'e list" shows "c_ipurge I D u (x # xs) = (if D x \<in> c_sources I D u (x # xs) then x # c_ipurge I D u xs else c_ipurge I D u xs)"
divsor_ex: fixes x :: "nat" assumes "\<not> Prime x"   and "Suc 0 < x" shows "\<exists>u>Suc 0. \<exists>v>Suc 0. u * v = x"
IH_1: fixes ds :: "bool list" assumes "branch ds T\<^sub>1" shows "path ds T\<^sub>1"
simps_2: fixes uw :: "'effect fs_set \<times> 'state"   and \<phi> :: "'pred" shows "EF uw \<turnstile>\<^sub>L \<phi> = False"
Basis_nonneg: fixes i :: "'b" assumes "i \<in> Basis" shows "(0::'b) \<le> i"
rep_eq: fixes x :: "'a regular"   and xa :: "'a regular" shows "(x \<le> xa) = (Rep_regular x \<le> Rep_regular xa)"
functor_axioms: shows "functor (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) G"
measure_spmf_return_spmf: fixes x :: "'a" shows "measure_spmf (return_spmf x) = return (count_space UNIV) x"
sums_Re: fixes X :: "nat \<Rightarrow> complex"   and a :: "complex" assumes "X sums a" shows "(\<lambda>n. Re (X n)) sums Re a"
prog_not_eq_in_ctran_aux: fixes P :: "'b com option"   and s :: "'b"   and Q :: "'b com option"   and t :: "'b" assumes "(P, s) -c\<rightarrow> (Q, t)" shows "P \<noteq> Q"
farkas_coefficients_ns: fixes ns :: "QDelta ns_constraint set" assumes "finite ns" shows "(\<exists>C. farkas_coefficients_ns ns C) = (\<nexists>v. v \<Turnstile>\<^sub>n\<^sub>s\<^sub>s  ns)"
set_nths: fixes xs :: "'g list"   and I :: "nat set" shows "set (nths xs I) = {xs ! i |i. i < length xs \<and> i \<in> I}"
asm: shows "\<forall>p. ps (S t i) p \<noteq> None"
rdvd_left1_int: fixes t :: "real" assumes "real_of_int \<lfloor>t\<rfloor> = t" shows "1 rdvd t"
disc_iff_1: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lnull (lappend xs ys) = (lnull xs \<and> lnull ys)"
S4b_live: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> (\<forall>l. $MemInv mm l)) \<and> WF(MemReturn rmCh ires p)_m p \<longrightarrow> (S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<leadsto> S5 rmhist p)"
trunc_bound_euclE: fixes p :: "nat"   and x :: "'a" obtains err :: "'a" where "\<bar>err\<bar> \<le> snd (trunc_bound_eucl p x)" and "fst (trunc_bound_eucl p x) = x + err"
assms_6: shows "A $ m $ mod_type_class.from_nat k \<noteq> (0::'a)"
is_arg_min_linorder: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "is_arg_min f P x = (P x \<and> (\<forall>y. P y \<longrightarrow> f x \<le> f y))"
IH: shows "\<forall>ys. length ys < length xs \<longrightarrow> distinct ys \<longrightarrow> (\<forall>x xa. (x, xa) = closest_pair_rec ys \<longrightarrow> distinct x)"
zero: fixes A :: "(('a, 'c) vec, 'b) vec" shows "A *v 0 = 0"
f6: shows "low_equal (snd (fst (get_curr_win () s1))) (snd (fst (get_curr_win () s2)))"
take_bit_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit 0 a = zero"
CASE'_3: shows "((ss, ce), e, ss', {#s'#} + cc) \<in> gtrs"
rlex_def: fixes xs :: "nat list"   and ys :: "nat list" shows "(xs <\<^sub>r\<^sub>l\<^sub>e\<^sub>x ys) = (rev xs <\<^sub>l\<^sub>e\<^sub>x rev ys)"
Call: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and p :: "'b"   and bdy :: "('a, 'b, 'c) com"   and s :: "'a" assumes "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile> (Call p, Normal s) \<rightarrow> (bdy, Normal s)"
mu_times_mu1: shows "mu * mu1 = monomial (punit.lc (q g1)) (punit.lt (q g1))"
distinct_merge: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "set xs \<inter> set ys = {}"   and "distinct xs"   and "distinct ys" shows "distinct (Common.merge f xs ys)"
finSeqSet: fixes S :: "sequent" shows "finite (NominalSequents.set_of_seq S)"
comp_point_mkArr': fixes a :: "'s"   and b :: "'s"   and F :: "'s \<Rightarrow> 's"   and x :: "'s" assumes "ide a"   and "ide b"   and "F \<in> hom unity a \<rightarrow> hom unity b"   and "\<guillemotleft>x : unity \<rightarrow> a\<guillemotright>" shows "mkArr' a b F \<cdot> x = F x"
eqNone: fixes t :: "'t" shows "(TS1 t = None) = (TS2 t = None)"
uniqify_mset: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.queue_invar q" shows "SkewBinomialHeapStruc.queue_to_multiset q = SkewBinomialHeapStruc.queue_to_multiset (SkewBinomialHeapStruc.uniqify q)"
ESem_subst: fixes e :: "exp"   and \<sigma> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and x :: "Vars.var"   and y :: "Vars.var" shows "\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<sigma>(x := \<sigma> y)\<^esub> = \<N>\<lbrakk> e[x::=y] \<rbrakk>\<^bsub>\<sigma>\<^esub>"
distinct_33: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and pi' :: "pi" shows "name1{name2}.pi \<noteq> !pi'"
split_sels_2: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and prod :: "'b \<times> 'c" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<not> (prod = (fst prod, snd prod) \<and> \<not> P (f (fst prod) (snd prod))))"
empty_correct_1: shows "rm.\<alpha> (rm.empty ()) = Map.empty"
iso_assoc: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src f = trg g"   and "src g = trg h" shows "B.iso \<a>[f, g, h]"
proj2_line_rep_def: fixes l :: "proj2_line" shows "proj2_line_rep l \<equiv> proj2_rep (L2P l)"
complex_of_real_add_Complex: fixes r :: "real"   and x :: "real"   and y :: "real" shows "cor r + Complex x y = Complex (r + x) y"
vimage_algebra_cong: fixes X :: "'b set"   and Y :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and M :: "'c measure"   and N :: "'c measure" assumes "X = Y"   and "\<And>x. x \<in> Y \<Longrightarrow> f x = g x"   and "sets M = sets N" shows "vimage_algebra X f M = vimage_algebra Y g N"
expand_superset: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "{a. g a \<noteq> (1::'b)} \<subseteq> A" shows "Prod_any g = prod g A"
RF_in_B: fixes x :: "'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and S :: "'k set" assumes "x \<in> B \<Gamma>" shows "(x \<in> RF\<^bsub>\<Gamma>\<^esub> S) = (x \<in> S)"
conv_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "f\<^sup>t = (\<lambda>(i, j). f (j, i)\<^sup>T)"
small: shows "small {\<gamma>. Limit \<gamma> \<and> \<gamma> \<subseteq>\<^sub>\<circ> \<alpha>}"
simps: fixes a1 :: "'c \<Rightarrow> 'd"   and a2 :: "('c, 'd) label list"   and a3 :: "'c \<Rightarrow> 'd" shows "ce_star a1 a2 a3 = ((\<exists>c. a1 = c \<and> a2 = [] \<and> a3 = c) \<or> (\<exists>c1 l c2 ls c3. a1 = c1 \<and> a2 = l # ls \<and> a3 = c3 \<and> ce c1 l c2 \<and> ce_star c2 ls c3))"
bounded_linear: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "bounded_linear f"
compactin_proper_map_preimage: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and K :: "'d set" assumes "proper_map X Y f"   and "compactin Y K" shows "compactin X {x \<in> topspace X. f x \<in> K}"
comm_neutral: fixes a :: "'m" shows "a + (0::'m) = a"
f2: shows "\<forall>b\<in>B. \<exists>a\<in>A. map f a \<cong> b"
s\<^sub>1_r: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Apply (VE (Lambda x e)) (VE v)])"
vars_for_consts_for_unis: fixes cs :: "'a list"   and p :: "('a, 'b) form" assumes "closed (length cs) p"   and "list_all (\<lambda>c. new c p) cs"   and "distinct cs" shows "vars_for_consts (consts_for_unis (put_unis (length cs) p) cs) cs = p"
rl: fixes a :: "'b option"   and b :: "'b" assumes "a = Some b" shows "the a = b"
closure_is_closed_csubspace: fixes S :: "'a set" assumes "csubspace S" shows "closed_csubspace (closure S)"
create_character_data_known_ptr: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list"   and new_character_data_ptr :: "(_) character_data_ptr" assumes "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>r new_character_data_ptr" shows "known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr)"
lhd_ldropn: fixes n :: "nat"   and xs :: "'b llist" assumes "enat n < llength xs" shows "lhd (ldropn n xs) = lnth xs n"
distrib: fixes g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "finite {a. g a \<noteq> (1::'b)}"   and "finite {a. h a \<noteq> (1::'b)}" shows "(\<Prod>a. g a * h a) = Prod_any g * Prod_any h"
permute_pure: fixes p :: "perm"   and x :: "'a" shows "p \<bullet> x = x"
sup_inf_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<sqinter> z \<squnion> x = y \<squnion> x \<sqinter> (z \<squnion> x)"
transfer_rule_numeral_uint: shows "rel_fun (=) cr_uint numeral numeral"
sphere_def: fixes x :: "'c"   and e :: "real" shows "sphere x e = {y. dist x y = e}"
da_elim_cases_40: fixes Env :: "env"   and B :: "lname set"   and v :: "Term.var"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (Acc v)\<guillemotright> A" obtains vn :: "lname" where "v = LVar vn" and "vn \<in> B" and "nrm A = B" and "brk A = (\<lambda>k. UNIV)"   | "\<forall>vn. v \<noteq> LVar vn" and "Env\<turnstile> B \<guillemotright>\<langle>v\<rangle>\<guillemotright> A"
monomial_is_monomial: fixes c :: "'c"   and t :: "'d" assumes "c \<noteq> (0::'c)" shows "is_monomial (monomial c t)"
char_Pg_le: fixes g :: "game"   and h :: "game" shows "(Abs_Pg (eq_game_rel `` {g}) \<le> Abs_Pg (eq_game_rel `` {h})) = ge_game (h, g)"
lookup_vectorize_poly: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and k :: "nat" shows "lookup (vectorize_poly p) k = proj_poly k p"
inset: shows "lset (ltake (enat (lfind P lst)) lst) \<subseteq> {x. (Not \<circ> P) x}"
fprod_cl2: shows "ideal (finprod (ideals_set R) I {..Suc n}) R"
monom_mult_unfold_2: fixes f :: "'a poly"   and n :: "nat" shows "f * Polynomial.monom (1::'a) n = monom_mult n f"
simps_1: fixes uu :: "'b" shows "is_mini_formula (Atom uu) = True"
seqE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.seq g f" obtains "VV.arr f" and "VV.arr g" and "VV.dom g = VV.cod f"
inner_axis': fixes i :: "'b"   and y :: "'a"   and x :: "('a, 'b) vec" shows "axis i y \<bullet> x = y \<bullet> x $ i"
res_fold_A_A_im_fold_A_A_im: shows "restrict1 fold_A (\<Union> A) \<turnstile> A = fold_A \<turnstile> A"
normal_times: fixes x :: "ordinal" assumes "0 < x" shows "normal ((*) x)"
maximalD2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and y :: "'b" assumes "class.order less_eq less"   and "x \<in> order.maximal less_eq f S"   and "y \<in> S" shows "less_eq (f y) (f x)"
scale_left_commute: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and x :: "'b" assumes "module scale" shows "scale a (scale b x) = scale b (scale a x)"
f1: shows "Deliver (i, Create i e) \<in> set (history j)"
opp_add: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "opp (local.add a p\<^sub>1 p\<^sub>2) = local.add a (opp p\<^sub>1) (opp p\<^sub>2)"
enabled_disj: fixes F :: "state \<times> state \<Rightarrow> bool"   and G :: "state \<times> state \<Rightarrow> bool" shows "\<turnstile> Enabled (F \<or> G) = (Enabled F \<or> Enabled G)"
iadd_0_right_enat: fixes a :: "enat" shows "a + enat 0 = a"
simf: shows "sim SPR.MC spr_simMC spr_sim"
Basis_list: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Basis_list :: "'c list"   and eucl_down :: "int \<Rightarrow> 'c \<Rightarrow> 'c"   and eucl_truncate_down :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and eucl_truncate_up :: "nat \<Rightarrow> 'c \<Rightarrow> 'c" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "set Basis_list = Basis"
s_has_real_derivative: fixes t :: "real" assumes "0 \<le> t"   and "v\<^sub>e / a\<^sub>e \<le> 0"   and "a\<^sub>e \<noteq> 0" shows "(ego2.s has_real_derivative ego2.s' t) (at t within {0..})"
Gromov_hyperbolic_invariant_under_quasi_isometry_explicit: fixes lambda :: "real"   and C :: "real"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry f" shows "Gromov_hyperbolic_subset (752 * lambda ^ 3 * (C + deltaG TYPE('c))) UNIV"
option_expand': fixes option :: "'c option"   and option' :: "'c option" assumes "(option = None) = (option' = None)"   and "\<And>x y. \<lbrakk>option = Some x; option' = Some y\<rbrakk> \<Longrightarrow> x = y" shows "option = option'"
numeral_uint: fixes n :: "num" shows "numeral n = Uint (numeral n)"
finsum_restrict: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "True"   and "restrict f A = restrict g A" shows "sum f A = sum g A"
2: fixes xa :: "(real \<times> real) list \<times> (real \<times> real) list"   and xb :: "(real \<times> real) list"   and y :: "(real \<times> real) list"   and xaa :: "real \<times> real" assumes "xa = partition (coll 0 x) xs"   and "(xb, y) = xa"   and "xaa = x + sum_list xb" shows "distinct (independent_pdevs y)"
fold_def: shows "DList_Set.fold \<equiv> map_fun id (map_fun list_of_dlist id) fold"
INST': fixes x :: "PDDL_STRIPS_Semantics.term"   and T :: "PDDL_STRIPS_Semantics.type" assumes "is_of_type (ty_term Q objT) x T" shows "is_of_type objT (subst_term f x) T"
simps_1: fixes f :: "'a" shows "EVAL \<^bold>\<langle>f\<^bold>\<rangle>\<^sub>0 = (if B.arr f then f else B.null)"
inject: fixes x21 :: "'b"   and x22 :: "'b llist"   and y21 :: "'b"   and y22 :: "'b llist" shows "(x21 ## x22 = y21 ## y22) = (x21 = y21 \<and> x22 = y22)"
empty_eq_union: fixes M :: "'k multiset"   and N :: "'k multiset" shows "({#} = M + N) = (M = {#} \<and> N = {#})"
snth_of_seq: fixes f :: "nat \<Rightarrow> 'b" shows "(!!) (of_seq f) = f"
A42_DAcc_level1: shows "DAcc level1 sA42 = {sA5}"
eventually_mp: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x"   and "eventually P F" shows "eventually Q F"
Sup_V_insert: fixes A :: "V set"   and x :: "V" assumes "small A" shows "\<Squnion> (insert x A) = x \<squnion> \<Squnion> A"
vimage_mono: fixes A :: "'e set"   and B :: "'e set"   and f :: "'f \<Rightarrow> 'e" assumes "A \<subseteq> B" shows "f -` A \<subseteq> f -` B"
set_take_subset: fixes n :: "nat"   and xs :: "'e list" shows "set (take n xs) \<subseteq> set xs"
coboundedI1: fixes c :: "'h"   and a :: "'h"   and b :: "'h" assumes "c \<le> a" shows "c \<le> max a b"
ptrq: shows "ptr = 2 ^ sz * word_of_nat q"
OUTfromVCorrect1_data5: shows "OUTfromVCorrect1 data5"
presWls_wlsInp: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'f, 'sort, 'g, 'h, 'c, 'i, 'j) Recursion.model_scheme" assumes "wlsInp delta inp"   and "presWls h MOD" shows "gWlsInp MOD delta (lift h inp)"
that: fixes x :: "'a" assumes "x \<in> s"   and "\<And>y. y \<in> s \<Longrightarrow> dist a x \<le> dist a y" shows "thesis"
leof_True_rule: fixes \<Phi> :: "'c \<Rightarrow> bool"   and m :: "'c nres" assumes "\<And>x. \<Phi> x" shows "m \<le>\<^sub>n SPEC \<Phi>"
nat_pow_mult: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat"   and m :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> (n + m)"
subseq_of_length_Nil: fixes n :: "nat"   and ys :: "'c list" shows "subseq_of_length n [] ys = (n = 0 \<and> ys = [])"
elims_1: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and y :: "bool" assumes "pred x xa = y" obtains f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = \<Up>f__" and "xa = cfa__ # cfsa__" and "y"   | Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = (Qa__)\<^sub>\<surd>" and "xa = cfb__ # cfsb__" and "y = Qa__ (fst cfb__)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and r :: "'ret" and p :: "'pname" and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Qb__:r\<hookrightarrow>\<^bsub>pa__\<^esub>fs" and "xa = cfc__ # cfsc__" and "y = Qb__ (fst cfc__, r)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and p :: "'pname" and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "y = (Q cf \<and> cfs \<noteq> [])"   | et :: "('var, 'val, 'ret, 'pname) edge_kind" where "x = et" and "xa = []" and "\<not> y"
relpowp_0_I: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" shows "(P ^^ 0) x x"
g_converg: fixes j :: "nat"   and v :: "nat" assumes "eval ph [j, v, i, e] \<down>" shows "eval g [j, v, i, e] \<down>= (if v < e_length e then v else if \<phi> i j \<down>= e_nth e j then v else j)"
add_increasing2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero c"   and "less_eq b a" shows "less_eq b (plus a c)"
Units_one_closed: shows "\<one>\<^bsub>K [X]\<^esub> \<in> Units (K [X])"
step: fixes y :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a" assumes "y \<in> directed_towards A r"   and "(x, y) \<in> r" shows "x \<in> directed_towards A r"
vec_simps_55: fixes aa :: "'a"   and x :: "'a"   and y :: "'a" shows "aa * (x - y) = aa * x - aa * y"
ex: shows "\<exists>q. q \<in> fst ` set polys \<and> poly (real_of_int_poly q) x = 0"
CptnModSeq2: fixes P0 :: "'b com"   and s :: "'b"   and xs :: "('b com option \<times> 'b) list"   and P1 :: "'b com"   and ys :: "('b com option \<times> 'b) list"   and zs :: "('b com option \<times> 'b) list" assumes "(Some P0, s) # xs \<in> cptn_mod"   and "fst (last ((Some P0, s) # xs)) = None"   and "(Some P1, snd (last ((Some P0, s) # xs))) # ys \<in> cptn_mod"   and "zs = map (lift P1) xs @ ys" shows "(Some (Seq P0 P1), s) # zs \<in> cptn_mod"
corthogonalD: fixes vs :: "'a vec list"   and i :: "nat"   and j :: "nat" assumes "corthogonal vs"   and "i < length vs"   and "j < length vs" shows "(vs ! i \<bullet>c vs ! j = (0::'a)) = (i \<noteq> j)"
strategy_proofI: fixes ds :: "'d set"   and \<phi> :: "('d \<Rightarrow> ('x \<times> 'x) set) \<Rightarrow> ('h \<Rightarrow> 'x set \<Rightarrow> 'x set) \<Rightarrow> 'd set \<Rightarrow> 'x set" assumes "\<And>Pd Pd' Ch d y. \<lbrakk>ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch; ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (Pd(d := Pd')) Ch; d \<in> ds; y \<in> \<phi> (Pd(d := Pd')) Ch ds; y \<in> Field (Pd d); \<forall>x\<in>dX (\<phi> Pd Ch ds) d. x \<noteq> y \<and> (x, y) \<in> Pd d\<rbrakk> \<Longrightarrow> False" shows "strategy_proof ds \<phi>"
strict_mono_on_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"
approx_indep_ge: fixes x :: "'b pmf"   and y :: "'b pmf"   and \<R> :: "('b pmf \<times> 'b pmf) set"   and \<alpha> :: "real"   and outcomes :: "'b set" assumes "x \<approx>[\<R>] y"   and "\<alpha> \<in> {0..1}"   and "rational_preference (lotteries_on outcomes) \<R>"   and "independent_vnm (lotteries_on outcomes) \<R>" shows "\<forall>r\<in>lotteries_on outcomes. mix_pmf \<alpha> y r \<succeq>[\<R>] mix_pmf \<alpha> x r"
cong: fixes prot :: "'a \<Rightarrow> bool"   and protFw :: "'a \<Rightarrow> bool"   and eintT :: "'a \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('b, 'a) efsym \<Rightarrow> univ list \<Rightarrow> univ" shows "ProblemIkTpart_TEModel.eenv prot protFw eintT eintF \<equiv> ProblemIkTpart_TEModel.eenv prot protFw eintT eintF"
simps_3: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env"   and es :: "(char list \<times> char list \<times> exp0) list"   and ux :: "char list" assumes "cakeml_static_env static_cenv" shows "cakeml_static_env.is_cupcake_value static_cenv (Recclosure env es ux) = (list_all (\<lambda>(_, uu_, y). is_cupcake_exp y) es \<and> cakeml_static_env.is_cupcake_all_env static_cenv env)"
finprod_one_eqI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> int" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = \<zero>" shows "finsum R f A = \<zero>"
is_operator_applicable_in_def: fixes s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator" shows "is_operator_applicable_in s op \<equiv> Let (precondition_of op) (list_all (\<lambda>v. s v = Some True))"
subset_eq: fixes A :: "'f set"   and B :: "'f set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
assms_2: fixes a :: "'b"   and b :: "'b" shows "f (a + b) = f a + f b"
hom_def: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d formula" shows "hom f x = (THE a. \<exists>S. hom_graph f S x a)"
disc_1: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "lnull xs \<or> \<not> P (lhd xs)" shows "lnull (ltakeWhile P xs)"