hcju/isabelleps huggingface dataset:
smaller_lits_are_false_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and t :: "'b trm"   and C :: "'b literal set"   and S :: "'b eclause set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.smaller_lits_are_false trm_ord sel t C S = (\<forall>L u v. (L \<in> C \<longrightarrow> basic_superposition.orient_lit trm_ord L u v pos \<longrightarrow> (u, t) \<in> trm_ord \<longrightarrow> (v, t) \<in> trm_ord \<longrightarrow> basic_superposition.trm_rep trm_ord sel u S \<noteq> basic_superposition.trm_rep trm_ord sel v S) \<and> (L \<in> C \<longrightarrow> basic_superposition.orient_lit trm_ord L u v neg \<longrightarrow> (u, t) \<in> trm_ord \<longrightarrow> (v, t) \<in> trm_ord \<longrightarrow> basic_superposition.trm_rep trm_ord sel u S = basic_superposition.trm_rep trm_ord sel v S))"
eq_mlex_I: fixes f :: "'b \<Rightarrow> nat"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>X Y. f X < f Y \<Longrightarrow> R X Y"   and "antisymp R" shows "{(X, Y). R X Y} = f <*mlex*> {(X, Y). f X = f Y \<and> R X Y}"
cases: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a stream"   and a2 :: "'b" assumes "transition_system.run execute enabled a1 a2" obtains a :: "'a" and p :: "'b" and r :: "'a stream" where "a1 = a ## r" and "a2 = p" and "enabled a p" and "transition_system.run execute enabled r (execute a p)"
s_UnauthV_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Unauth e\<ggreater> V\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Unauth e'a" and "\<lless>\<pi>, e\<ggreater> V\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"   | "\<pi> = e' # \<pi>'" and "e = Hash (hash e')" and "closed e'"
rel_mono: fixes R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R1a :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and R2a :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a" shows "Resumption.resumption.rel_resumption R1 R2 \<le> Resumption.resumption.rel_resumption R1a R2a"
conj_idem: fixes a :: "'b::refinement_lattice" shows "(a::'b::refinement_lattice) \<iinter> a = a"
elims: fixes x :: "le_rel"   and xa :: "le_rel"   and y :: "le_rel" assumes "x + xa = y" obtains "x = Leq_Rel" and "xa = Leq_Rel" and "y = Leq_Rel"   | uv_ :: "le_rel" where "x = Lt_Rel" and "xa = uv_" and "y = Lt_Rel"   | uu_ :: "le_rel" where "x = uu_" and "xa = Lt_Rel" and "y = Lt_Rel"
assms: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) list" assumes "ps \<in> set pss" shows "valid_decomp X ps"
imirror_eq_mirror_elem_image: fixes I :: "nat set" shows "imirror I = (\<lambda>x. mirror_elem x I) ` I"
SUP_bot: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>x. bot) ` A) = bot"
Cons_1: fixes us :: "'a vec list"   and us' :: "'a vec list" assumes "gram_schmidt_sub n us ws = us'"   and "set ws \<subseteq> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct (us @ ws)"   and "lin_indpt (set (us @ ws))"   and "corthogonal us" shows "set us' \<subseteq> carrier_vec n \<and> distinct us' \<and> corthogonal us' \<and> local.span (set (us @ ws)) = local.span (set us') \<and> length us' = length us + length ws"
upset_prop: fixes x :: "'c" shows "\<up> x = {y. x \<le> y}"
mn_arity: fixes n :: "nat"   and f :: "recf"   and a :: "abc_inst list"   and b :: "nat"   and c :: "nat" assumes "rec_ci (Mn n f) = (a, b, c)" shows "b = n"
HPhase0Read_HInv1: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HInv1 s"   and "HPhase0Read s s' p d" shows "HInv1 s'"
ASSUME_leof_iff: fixes \<Phi> :: "bool"   and \<Psi> :: "unit \<Rightarrow> bool" shows "(ASSUME \<Phi> \<le>\<^sub>n SPEC \<Psi>) = (\<Phi> \<longrightarrow> \<Psi> ())"
mod_0_less_imp_diff_Suc_div_conv: fixes n :: "nat"   and m :: "nat"   and r :: "nat" assumes "n mod m = 0"   and "r < m" shows "(n - Suc r) div m = n div m - Suc 0"
real_root_increasing: fixes n :: "nat"   and N :: "nat"   and x :: "real" assumes "0 < n"   and "n \<le> N"   and "0 \<le> x"   and "x \<le> 1" shows "root n x \<le> root N x"
bisim_simple_\<R>\<^sub>u: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "lc \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> lc'" shows "fst (fst lc) = fst (fst lc')"
kl: fixes e :: "'c" shows "pmf (map_pmf Inl Da) (Inr e) = 0"
unx_none_simps_2: fixes name :: "name" shows "unapp (free name) = None"
hyps: shows "p_power (1\<^sub>\<sigma> + x) (n + 1) = 1\<^sub>\<sigma> + x \<cdot> p_power (1\<^sub>\<sigma> + x) n"
P: fixes n :: "nat" assumes "n < length cl" shows "part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"
PLUS_def: fixes xs :: "'b language list" shows "PLUS xs \<equiv> foldr Plus xs Zero"
rel_sel: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "('d, 'f) optionT"   and b :: "('e, 'g) optionT" shows "rel_optionT' R1 R2 a b = R2 (run_option a) (run_option b)"
replicate_add: fixes n :: "nat"   and m :: "nat"   and x :: "'b" shows "x\<^bsup>n + m\<^esup> = x\<^bsup>n\<^esup> @ x\<^bsup>m\<^esup>"
measurable_bind2: fixes f :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and N :: "'b measure"   and g :: "'b \<Rightarrow> 'c measure"   and R :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "g \<in> N \<rightarrow>\<^sub>M subprob_algebra R" shows "(\<lambda>x. f x \<bind> g) \<in> M \<rightarrow>\<^sub>M subprob_algebra R"
build_F_def: fixes nds :: "('a, 'b) node_scheme set"   and \<phi> :: "'a ltlr" shows "build_F nds \<phi> \<equiv> (\<lambda>(\<mu>, \<eta>). name ` {q \<in> nds. \<mu> U\<^sub>r \<eta> \<in> old q \<longrightarrow> \<eta> \<in> old q}) ` until_frmlsr \<phi>"
term_brackets_Cons_SomeE: fixes v :: "nat"   and vs :: "nat list"   and t :: "dB"   and t' :: "dB" assumes "term_brackets (v # vs) t = Some t'" obtains s' :: "dB" where "term_brackets vs t = Some s'" and "term_bracket v s' = Some t'"
at_neg: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<forall>v'. v = c > v' \<longrightarrow> \<not> \<phi> ts v') = (\<not> (\<forall>v'. v = c > v' \<longrightarrow> \<phi> ts v'))"
GFP_fixed_point: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L"   and "Mono\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "GFP\<^bsub>L\<^esub> f \<in> fps L f"
fixp_lowerbound: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a" assumes "monotone orda orda f"   and "orda (f z) z" shows "orda (a.fixp f) z"
in_multiset_in_set: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in># mset xs) = (x \<in> set xs)"
evaln_SkipI: fixes G :: "prog"   and s :: "state"   and n :: "nat" shows "G\<turnstile>s \<midarrow>Skip\<midarrow>n\<rightarrow> s"
spec\<^sub>0_def: fixes p :: "prog" shows "spec\<^sub>0 p \<equiv> wfp p \<and> para p = [''x'', ''y''] \<and> (\<forall>x y. exec p [x, y] = Some (f x y)) \<and> costp p \<le> 3"
map_fun_apply: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'b"   and h :: "'e \<Rightarrow> 'f"   and x :: "'d" shows "map_fun f g h x = g (h (f x))"
sel_2: fixes x1 :: "'c match_expr"   and x2 :: "action" shows "get_action (Rule x1 x2) = x2"
Inter_iff: fixes A :: "'d"   and C :: "'d set set" shows "(A \<in> \<Inter> C) = (\<forall>X\<in>C. A \<in> X)"
reducedly_statically_complete_calculus_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" shows "reducedly_statically_complete_calculus Bot Inf entails Red_I Red_F \<equiv> calculus Bot Inf entails Red_I Red_F \<and> reducedly_statically_complete_calculus_axioms Bot Inf entails Red_I Red_F"
mk_monoid_simps_2: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and onea :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" assumes "class.comm_semiring_1 times onea plus zero" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
closure_eq_between: fixes S1 :: "'a set"   and S2 :: "'a set" assumes "convex S1"   and "convex S2" shows "(closure S1 = closure S2) = (rel_interior S1 \<subseteq> S2 \<and> S2 \<subseteq> closure S1)"
natural_extension_proj_def: shows "J.natural_extension_proj = (\<lambda>x. x 0)"
simps_3: fixes a :: "'c" shows "un_LCons2 (lconst a) = lconst a"
sources_aux_subset: fixes U :: "'a set"   and I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "U \<subseteq> sources_aux I D U xs"
monoI2: fixes X :: "nat \<Rightarrow> 'b" assumes "\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m" shows "monoseq X"
simps_2: fixes x :: "'e"   and xs :: "'e list" shows "rotate1 (x # xs) = xs @ [x]"
distinct_5: fixes x11 :: "bool"   and x12 :: "nat"   and x13 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x14 :: "nat"   and x15 :: "nat set"   and x16 :: "nat set"   and x17 :: "nat set"   and x18 :: "nat set"   and x41 :: "nat set"   and x42 :: "nat set"   and x43 :: "nat set"   and x44 :: "nat set" shows "Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 \<noteq> Ghost\<^sub>s\<^sub>b x41 x42 x43 x44"
relInvImage_UNIV_relImage: fixes R :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e" shows "R \<subseteq> BNF_Greatest_Fixpoint.relInvImage UNIV (BNF_Greatest_Fixpoint.relImage R f) f"
evt_def: fixes \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "\<diamond>\<^sub>b\<gamma> \<equiv> \<lambda>t n. \<exists>n'\<ge>n. \<gamma> t n'"
Rats_cases: fixes q :: "'d" assumes "q \<in> \<rat>" obtains r :: "rat" where "q = of_rat r"
iso_is_arr: fixes f :: "'a" assumes "Right.iso f" shows "Right.arr f"
fix_const: fixes c :: "'d" shows "(\<mu> x. c) = c"
size_remove1_mset_If: fixes M :: "'b multiset"   and x :: "'b" shows "size (remove1_mset x M) = size M - (if x \<in># M then 1 else 0)"
exM1_def: shows "exM1 = tensor_P (1\<^sub>m N) M1"
pr_strict'_def: fixes f :: "'f \<times> nat"   and g :: "'f \<times> nat" shows "pr_strict' f g = (f \<notin> lConsts \<and> (pr_strict f g \<or> g \<in> lConsts))"
mem_cball_0: fixes x :: "'b"   and e :: "real" shows "(x \<in> cball (0::'b) e) = (norm x \<le> e)"
brent: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and lambda :: "nat"   and mu :: "nat" assumes "properties f x0 lambda mu" shows "\<lbrace>\<lambda>s. True\<rbrace> fx0.brent f x0 \<lbrace>\<lambda>s. l s = lambda \<and> m s = mu\<rbrace>"
empty_replicate: fixes n :: "nat"   and x :: "'m" shows "([] = replicate n x) = (n = 0)"
get_owner_document_def: shows "get_owner_document \<equiv> l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document get_root_node get_disconnected_nodes"
of_rat_1_1: fixes x :: "rat" shows "invariant_1 (of_rat_1 x)"
get_disconnected_nodes_impl: shows "get_disconnected_nodes = local.a_get_disconnected_nodes"
Quotient3_rel: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs :: "'g \<Rightarrow> 'h"   and Rep :: "'h \<Rightarrow> 'g"   and r :: "'g"   and s :: "'g" assumes "Quotient3 R Abs Rep" shows "(R r r \<and> R s s \<and> Abs r = Abs s) = R r s"
ideal_add: fixes a :: "'a"   and A :: "'a set"   and b :: "'a"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "a \<in> A"   and "b \<in> A"   and "ideal A R add mult \<zero> \<one>" shows "add a b \<in> A"
is_of_type_def: fixes ty_ent :: "'b \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and v :: "'b"   and T :: "PDDL_STRIPS_Semantics.type" shows "is_of_type ty_ent v T = (case ty_ent v of None \<Rightarrow> False | Some vT \<Rightarrow> of_type vT T)"
decode_def: fixes M :: "int list"   and h :: "nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "decode M h prob \<equiv> if ast_problem.well_formed prob then if \<forall>op\<in>set (ast_problem.ast\<delta> prob). consistent_pres_op op then if \<forall>op\<in>set (ast_problem.ast\<delta> prob). is_standard_operator op then if dimacs_model M (SASP_to_DIMACS' h prob) then Inl (decode_DIMACS_model' M h prob) else Inr STR ''Error: Model does not solve the problem!'' else Inr STR ''Error: Conditional effects!'' else Inr STR ''Error: Preconditions inconsistent'' else Inr STR ''Error: Problem malformed!''"
commutes_def: fixes F :: "'b program"   and T :: "'b set"   and B :: "'b set"   and L :: "'b set set" shows "commutes F T B L \<equiv> \<forall>M. \<forall>act\<in>Acts F. B \<subseteq> M \<longrightarrow> cl L (T \<inter> wp act M) \<subseteq> T \<inter> (B \<union> wp act (cl L (T \<inter> M)))"
bigo_sum3: fixes f :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd"   and l :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and k :: "'e \<Rightarrow> 'f \<Rightarrow> 'c"   and A :: "'e \<Rightarrow> 'f set" assumes "f \<in> O(h)" shows "(\<lambda>x. \<Sum>y\<in>A x. l x y * f (k x y)) \<in> O(\<lambda>x. \<Sum>y\<in>A x. \<bar>l x y * h (k x y)\<bar>)"
Max_exact_aux_ge: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp m qs" shows "Max (poly_deg ` fst ` set qs) \<le> Max (poly_deg ` fst ` set (exact_aux k m qs))"
except_min_def: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and list_of_oalist :: "'c \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and oalist_of_list :: "('b \<times> 'd) list \<times> 'a \<Rightarrow> 'c"   and ko :: "'a"   and xs :: "'c" assumes "oalist_abstract rep_key_order list_of_oalist oalist_of_list" shows "oalist_abstract.except_min rep_key_order list_of_oalist oalist_of_list ko xs = oalist_abstract.tl list_of_oalist oalist_of_list (oalist_abstract.reorder rep_key_order list_of_oalist oalist_of_list ko xs)"
leg_simps_1: shows "C.arr assoc\<mu>\<nu>\<pi>.cod.leg0"
abs_eq: fixes x :: "'a" shows "singleton_pdevs x = Abs_pdev (\<lambda>i. if i = 0 then x else (0::'a))"
distr_convolution_ring_count_space: fixes M :: "('a \<times> 'a) measure"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "countable UNIV"   and "has_density M (count_space UNIV) f" shows "has_density (distr M (count_space UNIV) (\<lambda>(x, y). x + y)) (count_space UNIV) (\<lambda>z. \<integral>\<^sup>+ x. f (x, z - x) \<partial>count_space UNIV)"
smallomega_smallo_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<omega>[F](g)"   and "h \<in> o[F](g)" shows "f \<in> \<omega>[F](h)"
continuous_Re: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> complex" assumes "continuous F g" shows "continuous F (\<lambda>x. Re (g x))"
VPow_iff: fixes y :: "V"   and x :: "V" shows "(y \<in>\<^sub>\<circ> VPow x) = (y \<subseteq>\<^sub>\<circ> x)"
countable_basis_at_decseq: fixes x :: "'a" obtains A :: "nat \<Rightarrow> 'a set" where "\<And>i. open (A i)" and "\<And>i. x \<in> A i" and "\<And>S. \<lbrakk>open S; x \<in> S\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F i in sequentially. A i \<subseteq> S"
length_replicate: fixes n :: "nat"   and x :: "'k" shows "length (replicate n x) = n"
ne: shows "1 + x\<^sup>2 \<noteq> 0"
IH: assumes "1 \<le> n" shows "fact n + k \<le> fact (n + k)"
invariant_1_realI: fixes plr :: "int poly \<times> rat \<times> rat"   and x :: "real" assumes "root_cond plr x"   and "sgn (rai_lb plr) = sgn (rai_ub plr)"   and "\<exists>!x. root_cond plr x"   and "poly_cond (poly_real_alg_1 plr)" shows "invariant_1 plr \<and> real_of_1 plr = x"
out_var_plus: fixes x :: "'e \<Longrightarrow> 'h"   and y :: "'f \<Longrightarrow> 'h" shows "$(x +\<^sub>L y)\<acute> = $x\<acute> +\<^sub>L $y\<acute>"
simps_1: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre SKIP Q = Q"
prod_eq_def: fixes eqa :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and eqb :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x1 :: "'c \<times> 'e"   and x2 :: "'d \<times> 'f" shows "prod_eq eqa eqb x1 x2 \<equiv> case x1 of (a1, b1) \<Rightarrow> case x2 of (a2, b2) \<Rightarrow> eqa a1 a2 \<and> eqb b1 b2"
Inf_inf: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "\<forall>X. f (\<Sqinter>X) = \<Sqinter>{f x |x. x \<in> X}" shows "f (x \<sqinter> y) = f x \<sqinter> f y"
restrict_one: fixes ks :: "'c list"   and k :: "'c" assumes "\<not> List.member ks k" shows "(k # ks) \<langle>mone\<rangle> (k # ks) = [k] \<langle>mone\<rangle> [k] \<oplus> ks \<langle>mone\<rangle> ks"
godel_decode: fixes i :: "nat"   and nl :: "nat list" assumes "i < length nl" shows "Entry (godel_code nl) i = nl ! i"
LeftDerivation_Derives1_last: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "LeftDerivation a D b"   and "D \<noteq> []" shows "Derives1 (Derive a (take (length D - 1) D)) (fst (last D)) (snd (last D)) b"
0: shows "mat (row_length M1) (length M1) M1 \<and> mat (row_length M2) (length M2) M2 \<and> mat (row_length M3) (length M3) M3"
diff_invariant_def: fixes I :: "'b \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and G :: "'b \<Rightarrow> bool" shows "diff_invariant I f U S t\<^sub>0 G \<equiv> (\<mu> \<circ> \<P> x\<acute>=f & G on U S @ t\<^sub>0) {s. I s} \<subseteq> {s. I s}"
Inf_aform'_le: fixes e'' :: "nat \<Rightarrow> real"   and p :: "nat"   and r :: "'a \<times> 'a pdevs" assumes "\<And>i. e'' i \<le> 1"   and "\<And>i. - 1 \<le> e'' i" shows "Inf_aform' p r \<le> aform_val e'' r"
nadds_red_rtrancl_nonzero: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>q. q \<in> B \<Longrightarrow> \<not> lt q adds\<^sub>t lt p"   and "p \<noteq> 0"   and "(red B)\<^sup>*\<^sup>* p r" shows "r \<noteq> 0"
assms_3: shows "b1 \<inter> b2 \<noteq> {}"
that: fixes c :: "real" assumes "c \<in> {a..b}"   and "integral {a..b} (\<lambda>x. g x * f x) = g a * integral {a..c} f + g b * integral {c..b} f" shows "thesis"
iso_tuple_surjective_proof_assist_def: fixes x :: "'d"   and y :: "'e"   and f :: "'d \<Rightarrow> 'e" shows "Record.iso_tuple_surjective_proof_assist x y f = (f x = y)"
RETURN_refine_iff: fixes x :: "'e"   and R :: "('e \<times> 'f) set"   and y :: "'f" shows "(RETURN x \<le> \<Down> R (RETURN y)) = ((x, y) \<in> R)"
stable_on_CH: fixes ds :: "'b set"   and X :: "'a set" assumes "PdXXX.stable_on ds X" shows "PdXXX.CH X = X"
degree_le: fixes d :: "nat"   and x :: "'b pdevs" assumes "\<forall>j\<ge>d. pdevs_apply x j = (0::'b)" shows "degree x \<le> d"
exhaust: fixes y :: "'m \<times> 'n" obtains a :: "'m" and b :: "'n" where "y = (a, b)"
lins2D_4: fixes j :: "nat"   and k :: "nat"   and e :: "node fset \<times> command_gadget \<times> node fset"   and w :: "node" assumes "j < length (Inl v # \<pi>)"   and "k < length (Inl v # \<pi>)"   and "(Inl v # \<pi>) ! j = Inr e"   and "(Inl v # \<pi>) ! k = Inl w"   and "w |\<in>| thd3 e" shows "j < k"
good_item_simps_44: fixes rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xsa :: "'varSort"   and xa :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "goodEnv rhoa"   and "rhoa xsa xa = Some X" shows "Var xsa xa #[rhoa] = X"
Proj1_pmf_def: fixes D :: "('e + 'f) pmf" shows "Proj1_pmf D = map_pmf (\<lambda>a. case a of Inl e \<Rightarrow> e) (cond_pmf D {f. \<exists>e. Inl e = f})"
higher_deriv_const: fixes n :: "nat"   and c :: "'b" shows "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n = 0 then c else (0::'b))"
idem: fixes a :: "'a" shows "a \<^bold>\<squnion> a = a"
span_diff: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and S :: "'d set"   and y :: "'d" assumes "module scale"   and "x \<in> module.span scale S"   and "y \<in> module.span scale S" shows "x - y \<in> module.span scale S"
DBM_up_complete': fixes v :: "'e \<Rightarrow> nat"   and u :: "'e \<Rightarrow> 'f"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and n :: "nat" assumes "\<forall>c. 0 < v c"   and "u \<in> [M]\<^bsub>v,n\<^esub>\<^sup>\<up>" shows "u \<in> [up M]\<^bsub>v,n\<^esub>"
seg_set_impl_aux: fixes l :: "nat"   and u :: "nat" assumes "l < u"   and "u \<le> length S"   and "distinct S" shows "seg_set_impl l u \<le> SPEC (\<lambda>r. r = {S ! j |j. l \<le> j \<and> j < u})"
convex_hull_eq: fixes s :: "'c set" shows "(convex hull s = s) = convex s"
ide_char: fixes a :: "'a" shows "Left.ide a = (Left.arr a \<and> ide a)"
left: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and x :: "'d"   and y :: "'e" assumes "galois_connection G"   and "x \<in> carrier \<X>\<^bsub>G\<^esub>"   and "y \<in> carrier \<Y>\<^bsub>G\<^esub>"   and "x \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>G\<^esub> y" shows "\<pi>\<^sup>*\<^bsub>G\<^esub> x \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y"
mod_hom_def: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and N :: "('h, 'l, 'm) module_scheme"   and f :: "'j \<Rightarrow> 'l" shows "mod_hom R M N f \<equiv> module R M \<and> module R N \<and> mod_hom_axioms R M N f"
dual_def: fixes po :: "'b potype" shows "dual po \<equiv> \<lparr>pset = pset po, order = (order po)\<inverse>\<rparr>"
vector_scaleR_component: fixes r :: "real"   and x :: "('d, 'e) vec"   and i :: "'e" shows "(r *\<^sub>R x) $ i = r *\<^sub>R x $ i"
pi_approx_54_decimals: shows "\<bar>pi - 3141592653589793238462643383279502884197169399375105821 / 10 ^ 54\<bar> \<le> inverse (10 ^ 54)"
simCasesCont: fixes Rel :: "(pi \<times> pi) set"   and C :: "'a"   and Q :: "pi"   and P :: "pi" assumes "eqvt Rel"   and "\<And>Q' a x. \<lbrakk>x \<sharp> C; Q \<longmapsto> a<\<nu>x> \<prec> Q'\<rbrakk> \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"   and "\<And>Q' a x. \<lbrakk>x \<sharp> C; Q \<longmapsto> a<x> \<prec> Q'\<rbrakk> \<Longrightarrow> \<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel"   and "\<And>Q' \<alpha>. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>\<^sup>^<Rel> Q"
execute_new: fixes n :: "nat"   and x :: "'b"   and h :: "heap" shows "execute (Array.new n x) h = Some (Array.alloc (replicate n x) h)"