hcju/isabelleps huggingface dataset:
ind_topology_axioms_def: fixes X :: "'a set"   and S :: "'a set" shows "ind_topology_axioms X S \<equiv> S \<subseteq> X"
sum_sum_eq_sum_Un: fixes A :: "nat \<Rightarrow> nat set" assumes "\<forall>x\<in>{1..m}. \<forall>y\<in>{1..m}. x \<noteq> y \<longrightarrow> A x \<inter> A y = {}"   and "\<forall>x\<in>{1..m}. finite (A x)" shows "(\<Sum>x = 1..m. sum t (A x)) = sum t (\<Union> (A ` {1..m}))"
coeff_mult: fixes p :: "'a poly"   and q :: "'a poly"   and n :: "nat" shows "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n - i))"
string_of_nat_def: shows "string_of_nat \<equiv> inv nat_of_string"
my_\<R>\<^sub>3_aux_induct: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and Q :: "('Var, 'AExp, 'BExp) Stmt \<Rightarrow> (Mode \<Rightarrow> 'Var set) \<Rightarrow> ('Var \<Rightarrow> 'Val) \<Rightarrow> ('Var \<Rightarrow> 'BExp set option) \<Rightarrow> 'Var set \<times> 'Var set \<Rightarrow> 'BExp set \<Rightarrow> ('Var, 'AExp, 'BExp) Stmt \<Rightarrow> (Mode \<Rightarrow> 'Var set) \<Rightarrow> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<And>c\<^sub>1 mds mem\<^sub>1 \<Gamma> \<S> P c\<^sub>2 mem\<^sub>2 c \<Gamma>' \<S>' P'. \<lbrakk>\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; \<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'\<rbrakk> \<Longrightarrow> Q (c\<^sub>1 ;; c) mds mem\<^sub>1 \<Gamma>' \<S>' P' (c\<^sub>2 ;; c) mds mem\<^sub>2"   and "\<And>c\<^sub>1 mds mem\<^sub>1 \<Gamma> \<S> P c\<^sub>2 mem\<^sub>2 c \<Gamma>' \<S>' P'. \<lbrakk>\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; Q c\<^sub>1 mds mem\<^sub>1 \<Gamma> \<S> P c\<^sub>2 mds mem\<^sub>2; \<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'\<rbrakk> \<Longrightarrow> Q (c\<^sub>1 ;; c) mds mem\<^sub>1 \<Gamma>' \<S>' P' (c\<^sub>2 ;; c) mds mem\<^sub>2" shows "Q c\<^sub>1 mds mem\<^sub>1 \<Gamma> \<S> P c\<^sub>2 mds mem\<^sub>2"
D_def: shows "D = {0<..<1} \<times> {0<..<1}"
zero_in_smallo: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "(\<lambda>_. 0::'c) \<in> o[F](f)"
csn': shows "cs\<^bsup>\<rho>\<^esup> n = cs\<^bsup>\<rho>'\<^esup> n'"
sum_mat_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b mat" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum_mat plus zero A = comm_monoid_add.sum plus zero (($$) A) ({0..<dim_row A} \<times> {0..<dim_col A})"
abs_absolutely_integrableI: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "f integrable_on S"   and "(\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) integrable_on S" shows "f absolutely_integrable_on S"
simps_20: fixes v :: "char list"   and va :: "'a exp"   and vb :: "char list list"   and vc :: "'a exp" shows "val_of (RI (v,va) ; vb \<leftarrow> vc) = None"
cobounded1: fixes a :: "'d"   and b :: "'d" shows "inf_class.inf a b \<le> a"
tagged_partial_division_of_Union_self: fixes p :: "('a \<times> 'a set) set"   and s :: "'a set" assumes "p tagged_partial_division_of s" shows "p tagged_division_of \<Union> (snd ` p)"
fundamental_theorem_of_linear_inequalities_C_imp_B_full_dim: fixes A :: "'a vec set"   and b :: "'a vec" assumes "A \<subseteq> carrier_vec n"   and "finite A"   and "local.span A = carrier_vec n"   and "b \<in> carrier_vec n"   and "\<nexists>c B. B \<subseteq> A \<and> c \<in> {normal_vector B, - normal_vector B} \<and> Suc (card B) = n \<and> lin_indpt B \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)" shows "\<exists>B\<subseteq>A. lin_indpt B \<and> card B = n \<and> b \<in> finite_cone B"
utility_le_iff: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "vnm_utility carrier le u"   and "x \<in> carrier"   and "y \<in> carrier" shows "(u x \<le> u y) = le x y"
pa_cond_def: fixes S :: "('a \<times> 'b, unit) HashMap.hashmap \<times> ('a \<times> 'b) list \<times> ('a \<times> 'b, 'c) ta_rule dlist" shows "pa_cond S \<equiv> let (Q, W, \<delta>d) = S in W \<noteq> []"
Sup_pres_ladj_aux: fixes f :: "'b \<Rightarrow> 'c" assumes "Sup_pres f" shows "f \<stileturn> radj f"
invertible_coprime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo (times a b) c = one" shows "algebraic_semidom.coprime times one a c"
l_comb_Suc: fixes H :: "'a set"   and A :: "'b set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "H \<subseteq> carrier M"   and "ideal R A"   and "s \<in> {j. j \<le> Suc n} \<rightarrow> carrier R"   and "f \<in> {j. j \<le> Suc n} \<rightarrow> H" shows "l_comb R M (Suc n) s f = l_comb R M n s f \<plusminus> s (Suc n) \<cdot>\<^sub>s f (Suc n)"
lenext_def: fixes gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" shows "lenext gts ys xs = (length xs < length ys \<or> length ys = length xs \<and> gts ys xs)"
wf_cdecl_supD: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list"   and D :: "char list"   and r :: "fdecl list \<times> 'a mdecl list" assumes "wf_cdecl wf_md P (C, D, r)"   and "C \<noteq> Object" shows "is_class P D"
partition_on_space: fixes A :: "'b set" assumes "A \<noteq> {}" shows "partition_on A {A}"
equiv_iff_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(x \<simeq> y) = (x = y)"
a_def: shows "a \<equiv> \<lambda>k m. 1 / of_nat k * (\<Sum>d | d dvd gcd m k. g d * f (k div d) * of_nat d)"
simps_2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and err :: "'a"   and x :: "'b"   and xs :: "'b list"   and y :: "'c"   and ys :: "'c list"   and init :: "'a" shows "Matching.fold2 f err (x # xs) (y # ys) init = Matching.fold2 f err xs ys (f x y init)"
scff: shows "scf (f, length (bef @ s # aft)) (length bef) = Suc n"
Times_subset_cancel2: fixes x :: "'e"   and C :: "'e set"   and A :: "'f set"   and B :: "'f set" assumes "x \<in> C" shows "(A \<times> C \<subseteq> B \<times> C) = (A \<subseteq> B)"
circ_decompose_5: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> = (y\<^sup>\<star> * x\<^sup>\<star>)\<^sup>\<star>"
are_isomorphic: shows "\<exists>\<Phi>. invertible_functor S S' \<Phi> \<and> (\<forall>m. S.incl m \<longrightarrow> S'.incl (\<Phi> m))"
pred_set: fixes P :: "'c \<Rightarrow> bool" shows "pred_pmf P = (\<lambda>x. Ball (set_pmf x) P)"
perm_rel_lemma: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and \<pi> :: "perm"   and x :: "'b"   and y :: "'c" assumes "\<And>\<pi> x y. r (\<pi> \<bullet> x) (\<pi> \<bullet> y) \<Longrightarrow> r x y" shows "r (\<pi> \<bullet> x) (\<pi> \<bullet> y) = r x y"
Lim_def: fixes "open" :: "'b set \<Rightarrow> bool"   and A :: "'c filter"   and f :: "'c \<Rightarrow> 'b" assumes "class.t2_space open" shows "t2_space.Lim open A f = (THE l. filterlim f (topological_space.nhds open l) A)"
homogeneous_idealE_homogeneous: fixes F :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "p \<in> ideal F"   and "homogeneous p" obtains F' :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" and q :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" where "finite F'" and "F' \<subseteq> F" and "p = (\<Sum>f\<in>F'. q f * f)" and "\<And>f. homogeneous (q f)" and "\<And>f. f \<in> F' \<Longrightarrow> poly_deg (q f * f) = poly_deg p" and "\<And>f. f \<notin> F' \<Longrightarrow> q f = 0"
aval'_sound: fixes s :: "char list \<Rightarrow> int"   and S :: "char list \<Rightarrow> 'av"   and a :: "aexp" assumes "s \<in> \<gamma>\<^sub>f S" shows "aval a s \<in> \<gamma> (aval' a S)"
assms_22: shows "InvariantClCurrentLevel (getCl state) (getM state)"
list_all2_append2: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list"   and zs :: "'e list" shows "list_all2 P xs (ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> list_all2 P us ys \<and> list_all2 P vs zs)"
homeomorphism_injective_closed_map: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "f ` S = T"   and "inj_on f S"   and "\<And>U. closedin (top_of_set S) U \<Longrightarrow> closedin (top_of_set T) (f ` U)" obtains g :: "'b \<Rightarrow> 'a" where "homeomorphism S T f g"
A1: fixes n' :: "nat"   and x' :: "nat" assumes "((n', x'), n, x) \<in> lex_p" shows "c_assoc_have_key (pr_gr (loc_upb n' x')) (c_pair n' x') = 0"
sorted1: fixes x :: "'a word" shows "signed.sorted [x] = True"
q_continuous: fixes T :: "real set"   and v :: "real"   and s0 :: "real" shows "continuous_on T (movement.q v s0)"
primitive_def: fixes p :: "'b mpoly" shows "MPoly_Type.primitive p = (Gcd (MPoly_Type.coeffs p) = (1::'b))"
oth_class_taut_3_b: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>& \<psi> \<^bold>\<equiv> \<psi> \<^bold>& \<phi> in v]"
my_U_represents_pref: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "local.some_best \<succeq>[\<R>] local.some_worst \<and> (local.some_worst, local.some_best) \<notin> \<R>"   and "p \<in> local.\<P>"   and "q \<in> local.\<P>" shows "p \<succeq>[\<R>] q = (local.my_U q \<le> local.my_U p)"
terminate_F: fixes l :: "nat"   and lm :: "'a"   and tp :: "(action \<times> nat) list"   and stp :: "nat"   and m :: "nat"   and rs :: "nat"   and n :: "nat" assumes "steps0 (Suc 0, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"   and "tm_wf0 tp"   and "0 < rs" shows "terminate rec_F [code tp, bl2wc (<lm>)]"
ord_spmf_map_spmf2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a spmf"   and f :: "'a \<Rightarrow> 'a"   and q :: "'a spmf" shows "ord_spmf R p (map_spmf f q) = ord_spmf (\<lambda>x y. R x (f y)) p q"
eq_fold: fixes A :: "'a set" shows "Gcd\<^sub>f\<^sub>i\<^sub>n A = (if finite A then Finite_Set.fold gcd (0::'a) A else (1::'a))"
lookupDef_looksup: fixes g :: "'g"   and n :: "'node"   and v :: "'var"   and v' :: "'var"   and n' :: "'node"   and def :: "Def" assumes "lookupDef g n v = (v', n', def)"   and "n \<in> set (\<alpha>n g)" shows "v' = v"
simps_2: fixes x :: "'g list"   and xs :: "'g list list" shows "concat (x # xs) = x @ concat xs"
permute_zero: fixes x :: "'a" shows "0 \<bullet> x = x"
assms_2: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
eq_cong_ax_def: shows "eq_cong_ax = Var X1 EQ Var X2 AND Var X3 EQ Var X4 IMP Var X1 EQ Var X3 IMP Var X2 EQ Var X4"
"3_3": shows "is_heap (T v1 l1 r1)"
norm_v_X: fixes v :: "(complex, 'n) Finite_Cartesian_Product.vec"   and \<alpha> :: "complex" assumes "eigen_vector cA v \<alpha>" shows "map_vector cmod v \<in> X"
spr_simVal: shows "sim_val SPR.MC spr_simMC spr_sim"
extension_4: shows "comm_group (G\<lparr>carrier := H\<rparr> Mod K)"
Sigma_msetD1: fixes a :: "'b"   and b :: "'c"   and A :: "'b multiset"   and B :: "'b \<Rightarrow> 'c multiset" assumes "(a, b) \<in># Sigma_mset A B" shows "a \<in># A"
in_set_left: fixes y :: "'b"   and l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "y \<in> set_of l" shows "y \<in> set_of (Node l x False r)"
Int_empty_left: fixes B :: "'c set" shows "{} \<inter> B = {}"
right_neutral: fixes a :: "'e" shows "a * (1::'e) = a"
aux_lemma3: fixes l :: "'a list"   and q :: "'a list" assumes "set l \<subseteq> K"   and "q \<in> carrier (K [X])" shows "UP.eval (map poly_of_const l) q = UP.eval (map poly_of_const (local.normalize l)) q"
has_parametrized_subprob_density_def: fixes M :: "'a measure"   and N :: "'a \<Rightarrow> 'b measure"   and R :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal" shows "has_parametrized_subprob_density M N R f \<equiv> (\<forall>x\<in>space M. has_subprob_density (N x) R (f x)) \<and> (\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M R)"
game23: fixes x :: "nat"   and y :: "nat" shows "game2 x y = game3 (\<lambda>_ _ _ x. x) x y"
sc_eq_box_box_3: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) \<^bold>& \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<psi> \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>)) in v]"
h1b: shows "\<forall>i. MPoly_Type.degree (Var x ^ i) x = i"
rel_funE: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and x :: "'b"   and y :: "'c" assumes "rel_fun A B f g"   and "A x y" obtains "B (f x) (g y)"
splitAt_simpB: fixes ram :: "'a"   and a :: "'a"   and b :: "'a list" assumes "ram \<noteq> a" shows "fst (splitAt ram (a # b)) = a # fst (splitAt ram b)"
min: fixes y :: "'t" assumes "y \<prec>\<^sub>t v" shows "y \<notin> {v. v \<prec>\<^sub>t u \<and> d (pp_of_term v) \<le> dgrad_max d \<and> component_of_term v < length fs \<and> \<not> is_sig_GB_in d G v}"
spr_simAbs_refl: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "spr_sim t \<in> spr_simAbs ec"
sub: fixes a :: "'a" assumes "a \<in> G" shows "a \<in> M"
less_kbo_subst: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "less_kbo s t" shows "less_kbo (subst_atm_abbrev s \<sigma>) (subst_atm_abbrev t \<sigma>)"
isin_def: fixes t :: "'b tree"   and x :: "'b" shows "isin t x = is_root x (splay x t)"
1: shows "v \<notin> f ` set (remdups_wrt_rev f xs vs)"
head_del_arc: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c" shows "head (pre_digraph.del_arc G a) = head G"
subpath_reversepath: fixes g :: "real \<Rightarrow> 'b" shows "subpath 1 0 g = reversepath g"
MStep_Exec2_Aux: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and kb :: "nat"   and l1 :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and hh :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" assumes "(M, l, s, kb, l1, t) \<in> MStep"   and "\<exists>k. (M, l1, t, k, hh, v) \<in> Exec" shows "\<exists>n. (M, l, s, n, hh, v) \<in> Exec"
emeasure_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and A :: "'c set" assumes "\<Omega> \<inter> space M \<in> sets M"   and "A \<subseteq> \<Omega>" shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"
tune: fixes a :: "nat list"   and b :: "nat list" shows "check' a b (generate' (Max (set b)) (Max (set a)) a b) = fast_filter a b"
match_3: fixes \<mu> :: "'a"   and \<tau> :: "'a"   and \<nu> :: "'a" assumes "\<mu> \<star> \<tau> \<noteq> null"   and "\<nu> \<star> \<mu> \<noteq> null" shows "(\<nu> \<star> \<mu>) \<star> \<tau> \<noteq> null"
l10_4_spec: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and P' :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ReflectL Bet Cong P P' A B" shows "Tarski_neutral_dimensionless.ReflectL Bet Cong P' P A B"
mono_Seq_sym: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "(S `;` P) \<sqsubseteq> (S `;` Q)"
Sup_aform'_Affine_ge: fixes x :: "'a"   and X :: "'a \<times> 'a pdevs"   and p :: "nat" assumes "x \<in> Affine X" shows "x \<le> Sup_aform' p X"
triangle_left_iff_right: shows "((\<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g) \<star> f) \<cdot> (f \<star> \<eta>) = f) = ((g \<star> \<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g)) \<cdot> (\<eta> \<star> g) = g)"
c\<^sub>2: shows "stack c\<^sub>2 = s # stack c\<^sub>1"
part_equivp_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "part_equivp R = ((\<exists>x. R x x) \<and> (\<forall>x y. R x y = (R x x \<and> R y y \<and> R x = R y)))"
is_Prim0_Src: fixes t :: "'a local.term" shows "is_Prim\<^sub>0 (Src t)"
elims: fixes x :: "nat"   and y :: "tm" assumes "HTuple x = y" obtains "x = 0" and "y = HPair Zero Zero"   | k :: "nat" where "x = Suc k" and "y = HPair Zero (HTuple k)"
mul_closed_set_def: fixes R :: "('c, 'd) Ring_scheme"   and S :: "'c set" shows "mul_closed_set R S = (S \<subseteq> carrier R \<and> (\<forall>s\<in>S. \<forall>t\<in>S. s \<cdot>\<^sub>r\<^bsub>R\<^esub> t \<in> S))"
partition_on_mset_count_eq: fixes A :: "'a multiset"   and P :: "'a multiset multiset"   and a :: "'a" assumes "partition_on_mset A P"   and "a \<in># A" shows "(\<Sum>x\<in>#P. count x a) = count A a"
split_list_last: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> ran\<^sub>l xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> ran\<^sub>l zs"
subgroup_img_is_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and I :: "'c set" assumes "group_hom G H h"   and "subgroup I G" shows "subgroup (h ` I) H"
has_integral_integrable: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and s :: "'a set" assumes "(f has_integral i) s" shows "f integrable_on s"
sources_dom: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "sources (local.dom \<mu>) = sources \<mu>"
empty_subt: fixes t :: "fterm" shows "t \<cdot>\<^sub>t \<epsilon> = t"
assms: shows "locally_compatible_l R1 R2 (A + B) (C + D) (E + F)"
prefix_to_wordinterval_def: fixes pfx :: "'b prefix_match" shows "prefix_to_wordinterval pfx \<equiv> WordInterval (pfxm_prefix pfx) (pfxm_prefix pfx || pfxm_mask pfx)"
the_inv_into_map_Some: fixes f :: "'a \<Rightarrow> 'b"   and V :: "'a set"   and x :: "'b"   and y :: "'a" assumes "inj_on f V" shows "(the_inv_into_map V f x = Some y) = (y \<in> V \<and> x = f y)"
rel_sum_simps_4: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a2 :: "'e"   and b2 :: "'f" shows "rel_sum R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
minus_fract: fixes a :: "'a"   and b :: "'a" shows "- Fraction_Field.Fract a b = Fraction_Field.Fract (- a) b"
isort_correct: fixes xs :: "'a list" shows "isort xs = sort xs"
bvars\<^sub>s\<^sub>s\<^sub>t_cons_subset: fixes A :: "('e, 'd) stateful_strand_step list"   and a :: "('e, 'd) stateful_strand_step" shows "bvars\<^sub>s\<^sub>s\<^sub>t A \<subseteq> bvars\<^sub>s\<^sub>s\<^sub>t (a # A)"
dom_res_union_distr: fixes A :: "'c set"   and B :: "'c set"   and m :: "('c, 'd) fmap" shows "(A \<union> B) \<lhd> m = A \<lhd> m ++\<^sub>f B \<lhd> m"
converse_Id_on: fixes A :: "'e set" shows "(Id_on A)\<inverse> = Id_on A"
c_def: shows "c = Lcm ((nat \<circ> snd \<circ> quotient_of \<circ> poly.coeff p') ` {..Polynomial.degree p'})"
empty_clause_in_Q_of_Liminf_state: assumes "chain (\<leadsto>) Sts"   and "fair_state_seq Sts"   and "{#} \<in> Liminf_llist (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts)" shows "{#} \<in> Q_of_state (Liminf_state Sts)"
clopen: fixes S :: "'b set" shows "(closed S \<and> open S) = (S = {} \<or> S = UNIV)"
assms_6: shows "sKS secret \<notin> specKeysSecrets PQ"
abs_diff_le_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and a :: "'b"   and r :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (abs (minus x a)) r = (less_eq (minus a r) x \<and> less_eq x (plus a r))"
goodstein_impl: fixes c :: "nat"   and n :: "nat" assumes "c \<noteq> 0" shows "goodstein c n = goodsteinO c (C2O (N2H (c + 1) n))"
other_steps_def: fixes U :: "('c \<Rightarrow> 'd) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and J :: "'c set" shows "other_steps U J \<equiv> \<forall>\<sigma> \<sigma>'. U \<sigma> \<sigma>' \<longrightarrow> (\<forall>j\<in>J. \<sigma>' j = \<sigma> j)"
irreflp_greater: shows "irreflp (\<lambda>a b. cmp a b = Greater)"
hoaret_complete: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
hcomp_arr_obj: fixes f :: "('a E.term, 'a) arr"   and a :: "('a E.term, 'a) arr" assumes "arr f"   and "obj a"   and "src f = a" shows "f \<star> a = f"
unit_simps_3: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>D (F.unit a) = F.map\<^sub>0 a"
rimp_path_def: fixes X :: "'a list set"   and Z :: "'a list set" shows "rimp_path X Z = \<Union> {Y. p_prod X Y \<subseteq> Z}"
kruskal3_subset: shows "kruskal3 \<le>\<^sub>n SPEC (\<lambda>T. distinct T \<and> set T \<subseteq> superE)"
ln_unique: fixes y :: "real"   and x :: "real" assumes "exp y = x" shows "ln x = y"
ABC: shows "\<forall>t\<in>set A. \<forall>s\<in>set B. \<Gamma> t = \<Gamma> s \<and> mgu t s \<noteq> None \<longrightarrow> Q t \<and> Q s \<or> R t \<and> R s"
vertical_composite_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'a \<Rightarrow> 'c"   and G :: "'a \<Rightarrow> 'c"   and H :: "'a \<Rightarrow> 'c"   and \<sigma> :: "'a \<Rightarrow> 'c"   and \<tau> :: "'a \<Rightarrow> 'c" shows "vertical_composite A B F G H \<sigma> \<tau> \<equiv> (category A \<and> category B \<and> functor A B F) \<and> (functor A B G \<and> functor A B H) \<and> natural_transformation A B F G \<sigma> \<and> natural_transformation A B G H \<tau>"