hcju/isabelleps huggingface dataset:
irreducibleI: fixes a :: "'a"   and G :: "('a, 'b) monoid_scheme" assumes "a \<notin> Units G"   and "\<And>b. \<lbrakk>b \<in> carrier G; Divisibility.properfactor G b a\<rbrakk> \<Longrightarrow> b \<in> Units G" shows "Divisibility.irreducible G a"
complex_mat_char_poly_factorizable: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "\<exists>as. char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1:]) \<and> length as = n"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "C.ide ((F \<circ> G) a)"
RModuleHom_axioms: shows "RModuleHom FG (\<cdot>) V (\<cdot>) T"
Pd'_linear: fixes d :: "'b" shows "Linear_order (Pd' d)"
transitionsPreserveInvariants: fixes stateA :: "State"   and stateB :: "State"   and decisionVars :: "nat set"   and F0 :: "Literal list list" assumes "transition stateA stateB decisionVars"   and "invariantsHoldInState stateA F0 decisionVars" shows "invariantsHoldInState stateB F0 decisionVars"
IH: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" assumes "enabled cf \<omega>" shows "enabled ((cf ## \<omega>) !! n) (sdrop n \<omega>)"
bal_imp_full: fixes t :: "'a tree23" assumes "bal t" shows "full (height t) t"
rev_append_invert: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "(xs @ ys = rev zs) = (zs = rev ys @ rev xs)"
trans: fixes x :: "'alt"   and y :: "'alt"   and z :: "'alt" assumes "R i x y"   and "R i y z" shows "R i x z"
hseqE: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.hseq \<nu> \<mu>" obtains "B.arr \<mu>" and "B.arr \<nu>" and "src\<^sub>B \<nu> = trg\<^sub>B \<mu>"
inorder_ins: fixes t :: "'b tree234"   and x :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "inorder (tree\<^sub>i (ins x t)) = ins_list x (inorder t)"
cond_action_oks_conv_set: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and cts :: "'c conditional_action list" shows "final_thread.cond_action_oks final s t cts = (\<forall>ct\<in>set cts. final_thread.cond_action_ok final s t ct)"
bijections_in_domain_and_range_permutation: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A = card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} \<in> {f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B"
iteratei_rule_insert_P: fixes m :: "'s"   and I :: "('k \<times> 'v) set \<Rightarrow> 'a \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and c :: "'a \<Rightarrow> bool"   and f :: "'k \<times> 'v \<Rightarrow> 'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "invar m"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; (k, v) \<in> map_to_set (\<alpha> m) - it; it \<subseteq> map_to_set (\<alpha> m); I it \<sigma>\<rbrakk> \<Longrightarrow> I (insert (k, v) it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (map_to_set (\<alpha> m)) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> map_to_set (\<alpha> m); it \<noteq> map_to_set (\<alpha> m); \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iteratei m c f \<sigma>0)"
poly_def: fixes p :: "'b poly"   and a :: "'b" shows "poly p a = horner_sum id a (coeffs p)"
rbt_comp_inter_with_key: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_inter_with_key c = ord.rbt_inter_with_key (lt_of_comp c)"
dom_2: shows "dom \<rho> = vars_term t \<union> dom \<sigma>"
MakeCatMapsTo: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" assumes "f maps\<^bsub>C\<^esub> X to Y" shows "f maps\<^bsub>MakeCat C\<^esub> X to Y"
right_neutral: fixes a :: "'a" shows "a \<^bold>* \<^bold>1 = a"
subcls_irrefl: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and D :: "char list" assumes "wf_prog wf_md P"   and "(C, D) \<in> (subcls1 P)\<^sup>+" shows "C \<noteq> D"
path_\<mu>_def: shows "path_\<mu> = (INF p\<in>{p. apath u p v}. ereal (awalk_cost f p))"
in_progressS_eq: fixes S :: "Runs.var set option"   and S' :: "Runs.var set" shows "in_progressS S S' = (S \<noteq> None \<and> (\<forall>x\<in>S'. in_progress S x))"
\<alpha>_def: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" shows "\<alpha> \<mu> \<nu> \<tau> \<equiv> if VVV.arr (\<mu>, \<nu>, \<tau>) then (\<mu> \<star> \<nu> \<star> \<tau>) \<cdot> \<a>[local.dom \<mu>, local.dom \<nu>, local.dom \<tau>] else null"
xs_def: shows "xs = vfsequence_of_vlist (map vfsequence_of_vlist ys)"
wuntil_def: fixes \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma>' \<WW>\<^sub>c \<gamma> \<equiv> \<gamma>' \<UU>\<^sub>c \<gamma> \<or>\<^sup>c \<box>\<^sub>c\<gamma>'"
\<psi>_dim_1: shows "\<psi> \<in> carrier_vec N"
summable_norm_cos: fixes x :: "'a" shows "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x ^ n))"
mult_minus1: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and z :: "'d" assumes "class.ring_1 one times plus zero minus uminus" shows "times (uminus one) z = uminus z"
tau_s: shows "\<tau> 1\<^sub>\<sigma> = (0::'a)"
drop: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m Drop], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
intro: fixes WAC :: "'a while_algo"   and \<alpha> :: "'a \<Rightarrow> 'b"   and WAA :: "'b while_algo" assumes "\<forall>s. s \<in> wa_invar WAC \<and> \<alpha> s \<in> wa_cond WAA \<longrightarrow> s \<in> wa_cond WAC" shows "wa_precise_refine_axioms WAC WAA \<alpha>"
card_multisets_covering_set: fixes A :: "'c set"   and k :: "nat" assumes "finite A"   and "card A \<le> k" shows "card {M. set_mset M = A \<and> size M = k} = k - 1 choose (k - card A)"
unit_simps_2: fixes a :: "'a" assumes "B.obj a" shows "src\<^sub>B \<i>\<^sub>B[a] = a"
split_conv: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and a :: "'e"   and b :: "'f" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
disc2_noDstPorts: shows "\<forall>a. \<not> disc2 (Dst_Ports a)"
setmult_subset_G: fixes H :: "'a set"   and K :: "'a set" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G" shows "H <#> K \<subseteq> carrier G"
CLCodomain_def: fixes T :: "(ZF, ZF) Axioms" shows "CLCodomain T \<equiv> TCodomain \<circ> ZF2m T"
semiring_0_ow_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "semiring_0_ow U plus zero times \<equiv> semiring_ow U plus times \<and> comm_monoid_add_ow U plus zero \<and> mult_zero_ow U times zero"
le_add_diff: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b" shows "c \<subseteq># b + c - a"
CAS_\<tau>execI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e'' :: "'addr expr1"   and v :: "'addr val" assumes "\<tau>exec_move ci P t e' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (e\<bullet>compareAndSwap(D\<bullet>F, e', e'')) h (stk @ [v], loc, length (compE2 e) + pc, xcp) (stk' @ [v], loc', length (compE2 e) + pc', xcp')"
SUP_identity_eq: fixes A :: "'b set" shows "(\<Squnion>x\<in>A. x) = \<Squnion>A"
B: shows "B = (if A $$ (i, i + 1) = (1::'a) \<and> A $$ (i, j) \<noteq> (0::'a) then add_col_sub_row (- A $$ (i, j)) (Suc i) j A else A)"
finite_ball_avoid: fixes S :: "'a set"   and X :: "'a set"   and p :: "'a" assumes "open S"   and "finite X"   and "p \<in> S" shows "\<exists>e>0. \<forall>w\<in>ball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)"
degree4_coeffs: fixes p :: "'a poly" assumes "degree p = 4" shows "\<exists>a b c d e. p = [:e, d, c, b, a:] \<and> a \<noteq> (0::'a)"
codomains_char: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "codomains f = (if arr f then {cod f} else {})"
elims: fixes x :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem"   and y :: "nat" assumes "BsSkewBinomialHeapStruc.level x = y" obtains uu_ :: "'a" and uv_ :: "'b" and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" where "x = BsSkewBinomialHeapStruc.Element uu_ uv_ q" and "y = Suc (BsSkewBinomialHeapStruc.queue_level q)"
list_conj_def: fixes fs :: "'c fm list" shows "list_conj fs = foldr and fs TrueF"
cases: fixes a1 :: "cmd"   and a2 :: "nat"   and a3 :: "cmd" assumes "labels a1 a2 a3" obtains c :: "cmd" where "a1 = ca__" and "a2 = 0" and "a3 = ca__"   | V :: "char list" and e :: "expr" where "a1 = V:=e" and "a2 = 1" and "a3 = Skip"   | c\<^sub>1 :: "cmd" and l :: "nat" and c :: "cmd" and c\<^sub>2 :: "cmd" where "a1 = c\<^sub>1__;; c\<^sub>2__" and "a2 = la__" and "a3 = cb__;; c\<^sub>2__" and "labels c\<^sub>1__ la__ cb__"   | c\<^sub>2 :: "cmd" and l :: "nat" and c :: "cmd" and c\<^sub>1 :: "cmd" where "a1 = c\<^sub>1'__;; c\<^sub>2'__" and "a2 = lb__ + #:c\<^sub>1'__" and "a3 = cc__" and "labels c\<^sub>2'__ lb__ cc__"   | c\<^sub>1 :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" and c\<^sub>2 :: "cmd" where "a1 = if (b__) c\<^sub>1''__ else c\<^sub>2''__" and "a2 = lc__ + 1" and "a3 = cd__" and "labels c\<^sub>1''__ lc__ cd__"   | c\<^sub>2 :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" and c\<^sub>1 :: "cmd" where "a1 = if (ba__) c\<^sub>1 else c\<^sub>2" and "a2 = ld__ + #:c\<^sub>1 + 1" and "a3 = ce__" and "labels c\<^sub>2 ld__ ce__"   | c' :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" where "a1 = while (bb__) c'a__" and "a2 = l + 2" and "a3 = c;; while (bb__) c'a__" and "labels c'a__ l c"   | b :: "expr" and c' :: "cmd" where "a1 = while (b) c'" and "a2 = 1" and "a3 = Skip"   | p :: "char list" and es :: "expr list" and rets :: "char list list" where "a1 = Call p es rets" and "a2 = 1" and "a3 = Skip"
split_along_ivls2_def: fixes M :: "nat"   and X :: "'c set"   and Y :: "'c set" shows "split_along_ivls2 M X Y = sets_of_coll X \<bind> (\<lambda>Xs. FORWEAK Xs (RETURN op_empty_coll) (\<lambda>X. split_intersecting Y (mk_coll X) \<bind> (\<lambda>(I, N). split_along_ivls M (mk_coll X) I)) (\<lambda>x y. RETURN (y \<union> x)) \<bind> (\<lambda>Rs. RETURN (REMDUPS Rs)))"
assms_7: shows "(B, C) \<in> ns_mul_ext ns s"
interval_integral_norm2: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a" assumes "interval_lebesgue_integrable lborel a b f" shows "norm (interval_lebesgue_integral lborel a b f) \<le> \<bar>LBINT t=a..b. norm (f t)\<bar>"
assms_6: shows "\<forall>k\<in>{i + 1..<h}. a i \<le> a k"
Abs_oalist_inverse: fixes y :: "('a \<times> 'b) list \<times> 'a key_order" assumes "y \<in> Collect ko.oalist_inv" shows "list_of_oalist (Abs_oalist y) = y"
mod_add_self2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a b) b = modulo a b"
fune_UNIV: fixes C :: "Nielson_VCG.acom" shows "fune (transl C) UNIV = UNIV"
load_tmps_read_tmps_distinct_axioms: shows "load_tmps_read_tmps_distinct ts"
assms: shows "\<downharpoonright>\<alpha>\<downharpoonleft> \<noteq> pi"
AR_imp_absolute_retract_UNIV: fixes S :: "'c set"   and S' :: "'d set" assumes "AR S"   and "S homeomorphic S'"   and "closed S'" shows "S' retract_of UNIV"
assms: shows "rho xs (x @xs[z1 \<and> z2]_zs) = None"
thunks_append: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "thunks (\<Delta> @ \<Gamma>) = thunks \<Delta> \<union> (thunks \<Gamma> - domA \<Delta>)"
d_gt: fixes x :: "real" assumes "t < x"   and "0 < \<bar>x - t\<bar>"   and "\<bar>x - t\<bar> < d_gt" shows "\<bar>(f x - f t) / (x - t) - D_gt\<bar> < 1"
add_le_same_cancel2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus a b) b = less_eq a zero"
DEaxiom_def: shows "DEaxiom = ([[EvolveODE (OSing vid1 (f1 fid1 vid1)) (p1 vid2 vid1)]]P pid1 \<leftrightarrow> ([[EvolveODE (OSing vid1 (f1 fid1 vid1)) (p1 vid2 vid1)]][[DiffAssign vid1 (f1 fid1 vid1)]]P pid1))"
that: fixes V :: "'b set"   and W :: "'b set" assumes "openin (top_of_set U) V"   and "closedin (top_of_set U) W"   and "S' \<subseteq> V"   and "V \<subseteq> W"   and "S' retract_of W" shows "thesis"
leg0_prj_in_hom: shows "\<guillemotleft>T.leg0 \<cdot> \<p>\<^sub>0[S.leg0, T.leg1] : S.leg0 \<down>\<down> T.leg1 \<rightarrow> T.src\<guillemotright>"
simps_2: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" shows "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
mgu_always_unifies: fixes M :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and N :: "('c, 'd) Term.term" assumes "Unifier \<theta> M N" shows "\<exists>\<delta>. mgu M N = Some \<delta>"
relH_D2_2: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap"   and a :: "nat"   and t :: "typerep" assumes "relH as h h'"   and "a \<in> as" shows "arrays h t a = arrays h' t a"
div_mult_self1_is_m: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "n * m div n = m"
filterlim_at_top_dense: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "filterlim f at_top F = (\<forall>Z. \<forall>\<^sub>F x in F. Z < f x)"
assms_3: shows "\<forall>ref\<in>deps oper. ref < oid"
I_in_Gen: shows "I \<subseteq> Gen (\<P> k u)"
vsv_appE: fixes a :: "V"   and b :: "V" assumes "xs\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> xs" obtains "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> xs"
empty_correct: shows "map_of_heap empty \<subseteq>\<^sub>m Map.empty"
valid_selector_def: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and selector :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option" shows "valid_selector Rs selector \<equiv> \<forall>G. (worklist G Rs \<noteq> {} \<longrightarrow> (\<exists>(R, f)\<in>UNIV. selector G = Some (R, f) \<and> valid_selection Rs G R f)) \<and> (worklist G Rs = {} \<longrightarrow> selector G = None)"
base_in_hom_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c" assumes "tabulation_data V H \<a> \<i> src trg r \<rho> f g" shows "partial_magma.in_hom V r r r"
assms_1: shows "A\<^sup>2 < B\<^sup>2"
ik\<^sub>s\<^sub>tD': fixes t :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "t \<in> trms\<^sub>s\<^sub>t S"
de_morgan_2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.il_semiring inf sup bot less_eq less top"   and "\<exists>y. sup x y = top \<and> inf x y = bot \<and> inf y x = bot"   and "\<exists>ya. sup y ya = top \<and> inf y ya = bot \<and> inf ya y = bot"   and "\<exists>ya. sup (sup x y) ya = top \<and> inf (sup x y) ya = bot \<and> inf ya (sup x y) = bot" shows "il_semiring.neg inf sup bot top (sup x y) = inf (il_semiring.neg inf sup bot top x) (il_semiring.neg inf sup bot top y)"
nres_order_simps_21: fixes x :: "'a"   and Y :: "'a set" shows "(return x \<le> RES Y) = (x \<in> Y)"
R_Suc: fixes M :: "nat" assumes "N \<le> M" shows "R (Suc M) = real_of_int (a M) * R M - real_of_int (B * b (Suc M)) / real_of_int (a (Suc M))"
appliedDecide_def: fixes stateA :: "State"   and stateB :: "State"   and decisionVars :: "nat set" shows "appliedDecide stateA stateB decisionVars \<equiv> \<exists>l. var l \<in> decisionVars \<and> \<not> l el elements (getM stateA) \<and> \<not> opposite l el elements (getM stateA) \<and> getF stateB = getF stateA \<and> getM stateB = getM stateA @ [(l, True)]"
igSubstIPresIGWls_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstIPresIGWls MOD \<equiv> \<forall>ys Y y s X. igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> igWls MOD s (igSubst MOD ys Y y X)"
J: shows "J \<subseteq> {0..<dim_col (P * A * Q)}"
STL4E_gen: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior"   and G :: "'b \<Rightarrow> bool" assumes "sigma \<Turnstile> \<box>F"   and "\<turnstile> Init F \<longrightarrow> Init G" shows "sigma \<Turnstile> \<box>G"
bounded_linear_right: fixes a :: "'a" shows "bounded_linear (h a)"
op_plus_is_pr: shows "(+) \<in> PrimRec2"
inner_scaleR_right: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and r :: "real"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (scaleR r y) = r * inner x y"
diff_Pair: fixes a :: "'b"   and b :: "'c"   and c :: "'b"   and d :: "'c" shows "(a, b) - (c, d) = (a - c, b - d)"
equal_refl: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.equal equal" shows "equal x x = True"
induct: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and x1 :: "char list"   and x2 :: "char list list"   and x3 :: "char list"   and x4 :: "ty list \<times> ty \<times> char list list \<times> expr"   and x5 :: "char list list"   and Pa :: "char list \<Rightarrow> char list list \<Rightarrow> char list \<Rightarrow> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> char list list \<Rightarrow> bool" assumes "P \<turnstile> (x1,x2) selects x3 = x4 via x5"   and "\<And>C M mthd Cs' Cs. P \<turnstile> C has least M = mthd via Cs' \<Longrightarrow> Pa C Cs M mthd Cs'"   and "\<And>C M Cs mthd Cs'. \<lbrakk>\<forall>mthd Cs'. \<not> P \<turnstile> C has least M = mthd via Cs'; P \<turnstile> (C, Cs) has overrider M = mthd via Cs'\<rbrakk> \<Longrightarrow> Pa C Cs M mthd Cs'" shows "Pa x1 x2 x3 x4 x5"
sub_sub_PA: fixes I' :: "nat set" shows "submatrix (submatrix (P * A) I UNIV) UNIV I' = submatrix (P * A) I I'"
star_not_empty: fixes A :: "'a list set" shows "star A \<noteq> {}"
e: shows "[(c, b)] \<bullet> [a].x = [a].([(c, b)] \<bullet> x)"
no_grey_refs_locs_def: shows "no_grey_refs_locs = black_heap_locs \<union> sweep_locs \<union> {mark_end}"
inj_on_def: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
yes: fixes xs :: "'b list"   and x :: "'b" assumes "index xs x < length xs" shows "xs ! index xs x = x"
Var: fixes xs :: "'varSort"   and x :: "'var" shows "phi (asSort xs) (Var xs x)"
coeff_map_poly_hom: fixes p :: "'a poly"   and i :: "nat" shows "poly.coeff (map_poly hom p) i = hom (poly.coeff p i)"
object_ptr_kinds_eq_h: shows "object_ptr_kinds h2 = object_ptr_kinds h |\<union>| {|cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^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_shadow_root_ptr|}"
zero_closed: shows "0\<^sub>v n \<in> carrier_vec n"
at_swap: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "at TYPE('a)" shows "swap (a, b) (swap (a, b) c) = c"
CONST_INTF_def: fixes c :: "'f"   and I :: "interface" shows "c ::\<^sub>i I \<equiv> True"
elts_multE: fixes z :: "V"   and x :: "V"   and y :: "V" assumes "z \<in> elts (x * y)" obtains u :: "V" and v :: "V" where "u \<in> elts x" and "v \<in> elts y" and "z = x * v + u"
inject_3: fixes x3 :: "bool"   and y3 :: "bool" shows "(IBool x3 = IBool y3) = (x3 = y3)"
distinct_5: shows "SizeOp \<noteq> CharactersOp"
sym_digraph_axioms_def: fixes G :: "('c, 'd) pre_digraph" shows "sym_digraph_axioms G \<equiv> symmetric G"
maxR_absorb2: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> r" shows "maxR x y = y"
irreducibleE: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.irreducible times one zero p" obtains "p \<noteq> zero" and "\<not> dvd.dvd times p one" and "\<And>a b. p = times a b \<Longrightarrow> dvd.dvd times a one \<or> dvd.dvd times b one"
irreflI: fixes R :: "('a \<times> 'a) set" assumes "\<And>a. (a, a) \<notin> R" shows "irrefl R"
B: shows "0 < \<epsilon> / (\<bar>B\<bar> + 1)"
set_step2: fixes s :: "'c list"   and r :: "'c"   and mf :: "nat"   and sws :: "nat list" shows "set (step s r (mf, sws)) = set s"
index_def: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set"   and i :: "'b" shows "index f s i \<equiv> {x \<in> s. f x = i}"
reduce_fps_poly_monic: fixes F :: "'b fps poly" assumes "monic F" shows "monic (reduce_fps_poly F)"
abs_eq_iff: fixes x :: "'c ltl"   and y :: "'c ltl" shows "(Abs x = Abs y) = x \<equiv>\<^sub>P y"
monoI: fixes f :: "('a, 'b) Term.term \<Rightarrow> 'c" assumes "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<le> f y" shows "term.mono f"
hyps: fixes a :: "'a" assumes "f' a = foldr f xs (f' b)"   and "set xs \<subseteq> s" shows "g' a = foldr g xs (g' b)"
exhaust: fixes y :: "trm" obtains x1 :: "nat" where "y = Var x1"   | x21 :: "POPLmark.type" and x22 :: "trm" where "y = (\<lambda>:x21. x22)"   | x31 :: "POPLmark.type" and x32 :: "trm" where "y = (\<lambda><:x31. x32)"   | x41 :: "trm" and x42 :: "trm" where "y = x41 \<bullet> x42"   | x51 :: "trm" and x52 :: "POPLmark.type" where "y = x51 \<bullet>\<^sub>\<tau> x52"
suminf_mult2: fixes f :: "nat \<Rightarrow> 'd"   and c :: "'d" assumes "summable f" shows "suminf f * c = (\<Sum>n. f n * c)"