hcju/isabelleps huggingface dataset:
class_add_xcpt_app: fixes i :: "instr"   and P :: "jvm_prog"   and pc :: "nat"   and mxs :: "nat"   and xt :: "ex_table"   and \<tau> :: "ty list \<times> ty err list"   and wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and cdec :: "jvm_method class" assumes "xcpt_app i P pc mxs xt \<tau>"   and "wf_prog wf_md P"   and "\<not> is_class P C" shows "xcpt_app i (class_add P (C, cdec)) pc mxs xt \<tau>"
ipresSubstAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'varSort, 'l, 'm, 'var, 'h, 'i, 'n) model_scheme" shows "ipresSubstAbs h hA MOD \<equiv> \<forall>ys Y y us s A. wls (asSort ys) Y \<and> wlsAbs (us, s) A \<longrightarrow> hA (A $[Y / y]_ys) = igSubstAbs MOD ys (h Y) y (hA A)"
cmp\<^sub>U\<^sub>P_ide_simp: fixes f :: "'a"   and g :: "'a" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "cmp\<^sub>U\<^sub>P (f, g) = MkArr (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>) \<^bold>\<langle>f \<star>\<^sub>B g\<^bold>\<rangle> (f \<star>\<^sub>B g)"
l8_20_1_R2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and P :: "'a"   and C' :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong P C' D"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A C' C"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B D C" shows "Tarski_neutral_dimensionless.Per Bet Cong B A P"
simps_3: fixes var :: "'a"   and f :: "'b \<Rightarrow> ('a, 'b) trm" shows "SIGT ($f var f) = {var} \<union> (\<Union>i. SIGT (f i))"
Ide_HcompNml: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Ide t"   and "E.Ide u"   and "E.Nml t"   and "E.Nml u"   and "E.Src t = E.Trg u" shows "E.Ide (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u)"
spec_def: shows "spec \<equiv> \<lambda>xs. holds init_config xs \<and> alw next xs"
fps_compose_uminus: fixes a :: "'a fps"   and c :: "'a fps" shows "- a oo c = - (a oo c)"
subseqs_powset: fixes xs :: "'a list" shows "set ` set (subseqs xs) = Pow (set xs)"
sas_plus_formalism_and_induced_strips_formalism_are_equally_expressive_i_c: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and k :: "nat" assumes "is_valid_problem_sas_plus \<Psi>" shows "map (sasp_op_to_strips \<Psi>) ` bounded_solution_set_sas_plus' \<Psi> k = bounded_solution_set_strips' (\<phi> \<Psi> ) k"
P_pow: fixes i :: "nat" shows "P ((f ^^ i) bot)"
full_rank_lin_indpt: fixes A :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc"   and "rank A = nc"   and "distinct (cols A)" shows "lin_indpt (set (cols A))"
eval_def: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'b"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.eval tCMP active cid t t' n \<gamma> \<equiv> (\<exists>i\<ge>n. active cid (t i)) \<and> \<gamma> (lnth (dynamic_component.proj tCMP active cid (inf_llist t) @\<^sub>l inf_llist t')) (the_enat (dynamic_component.nAct tCMP active cid (enat n) (inf_llist t))) \<or> (\<exists>i. active cid (t i)) \<and> \<not> (\<exists>i'\<ge>n. active cid (t i')) \<and> \<gamma> (lnth (dynamic_component.proj tCMP active cid (inf_llist t) @\<^sub>l inf_llist t')) (dynamic_component.cnf2bhv tCMP active cid t n) \<or> (\<nexists>i. active cid (t i)) \<and> \<gamma> (lnth (dynamic_component.proj tCMP active cid (inf_llist t) @\<^sub>l inf_llist t')) n"
mask128: shows "340282366920938463463374607431768211455 = mask 128"
lset_lappend_conv: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lset (lappend xs ys) = (if lfinite xs then lset xs \<union> lset ys else lset xs)"
assoc'_naturality: fixes f0 :: "'c local.term set"   and f1 :: "'c local.term set"   and f2 :: "'c local.term set" assumes "\<F>C.arr f0"   and "\<F>C.arr f1"   and "\<F>C.arr f2" shows "\<F>C.tensor (\<F>C.tensor f0 f1) f2 \<cdot> \<F>C.assoc' (\<F>C.dom f0) (\<F>C.dom f1) (\<F>C.dom f2) = \<F>C.assoc' (\<F>C.cod f0) (\<F>C.cod f1) (\<F>C.cod f2) \<cdot> \<F>C.tensor f0 (\<F>C.tensor f1 f2)"
hyperbola_method: fixes A :: "real"   and B :: "real"   and x :: "real"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "0 \<le> A"   and "0 \<le> B"   and "A * B = x" shows "sum_upto (dirichlet_prod f g) x = sum_upto (\<lambda>n. f n * sum_upto g (x / real n)) A + sum_upto (\<lambda>n. sum_upto f (x / real n) * g n) B - sum_upto f A * sum_upto g B"
9: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<or> x \<squnion> y \<noteq> y"
neq: fixes k :: "nat" assumes "\<forall>i<k. f i = g i"   and "P (f k) (g k)" shows "thesis"
elementary_empty: obtains p :: "'c set set" where "p division_of {}"
inject_3: fixes x1 :: "name"   and x2 :: "trm"   and y1 :: "name"   and y2 :: "trm" shows "(\<Lambda> x1 . x2 = \<Lambda> y1 . y2) = ([x1].x2 = [y1].y2)"
Gamma_reflection_complex': fixes z :: "complex" shows "Gamma z * Gamma (- z) = - complex_of_real pi / (z * sin (complex_of_real pi * z))"
INF_identity_eq: fixes A :: "('c \<times> 'd) set" shows "Inf_prod (\<P> (\<lambda>x. x) A) = Inf_prod A"
option_rel_Some1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b option" shows "rel_option A \<lfloor>x\<rfloor> y = (\<exists>y'. y = \<lfloor>y'\<rfloor> \<and> A x y')"
lucas_lehmer_necessary: fixes p :: "nat" assumes "mersenne_prime p" shows "2 ^ p - 1 dvd gen_lucas_lehmer_sequence 4 (p - 2)"
Im_interval_minus: fixes A :: "complex_interval"   and B :: "complex_interval" shows "Im_interval (A - B) = Im_interval A - Im_interval B"
igSubstAllIPresIGWlsAll_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igSubstAllIPresIGWlsAll MOD \<equiv> igSubstIPresIGWls MOD \<and> igSubstAbsIPresIGWlsAbs MOD"
simps: fixes S :: "'b literal set set" shows "Res S = (let R = res S \<union> S in if R = S then Some S else Res R)"
simps_2: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.funcs \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = funcs"
extended_distance_basepoint: fixes x :: "'c"   and y :: "'c" shows "extended_distance (basepoint x) (basepoint y) = dist x y"
Class_def: shows "Class = (\<lambda>a\<in>S. {b \<in> S. (b, a) \<in> Equivalence})"
elims_1: fixes x :: "('b, 'c) Combinators list"   and y :: "bool" assumes "wellformed_policy1 x = y" obtains "x = []" and "y"   | xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "y = (DenyAll \<notin> set xs)"
S_def: shows "S \<equiv> count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M)"
bit_mod_2_iff: fixes a :: "'a"   and n :: "nat" shows "bit (a mod (2::'a)) n = (n = 0 \<and> odd a)"
eq_refl: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "x = y" shows "less_eq_prod x y"
insert_remove: fixes A :: "'x set"   and x :: "'x" assumes "finite A" shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
lc_f: shows "lincomb f (set (cols A)) = 0\<^sub>v n"
phi_tendsto: fixes x :: "real" assumes "x \<in> {a..b}" shows "(\<phi> \<longlongrightarrow> \<phi> x) (at x within {a..b})"
necT: fixes F :: "'a \<Rightarrow> bool" assumes "\<turnstile> F" shows "\<turnstile> \<box>F"
uses'_code_correct: fixes g :: "'g" assumes "uninst_code.ssa.redundant (u_g g s1_uses) (p_g g s1_phis) g" shows "uninst.uses' ((usesOf \<circ>\<circ>\<circ> u_g) g s1_uses) (\<lambda>ga. Mapping.lookup (p_g g s1_phis ga)) g = uninst_code.uses'_code (u_g g s1_uses) (p_g g s1_phis) g (uninst_code.chooseNext' (u_g g s1_uses) (p_g g s1_phis) g)"
constant_is_Zp_continuous: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "is_Zp_continuous (\<cc>\<^bsub>Zp\<^esub> a)"
s_set_def: fixes R :: "('f, 'g) Ring_scheme"   and s :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set" shows "s_set R s A = {x. \<exists>r\<in>carrier R. \<exists>a\<in>A. x = r \<^sub>s\<cdot> a} \<union> A"
map_comp: fixes g :: "'h \<Rightarrow> 'g"   and f :: "'i \<Rightarrow> 'h"   and v :: "'i list" shows "map g (map f v) = map (g \<circ> f) v"
idiv_enat_enat: fixes a :: "nat"   and b :: "nat" shows "enat a div enat b = enat (a div b)"
tendsto_eval_fds_truncate: fixes f :: "'a fds"   and s :: "'a" assumes "fds_converges f s" shows "(\<lambda>N. eval_fds (fds_truncate N f) s) \<longlonglongrightarrow> eval_fds f s"
PREFIX_def: fixes r :: "'b rexp" shows "PREFIX r = REV (SUFFIX (REV r))"
am6_lem: fixes x :: "'a"   and y :: "'a" shows "a (x \<cdot> y) \<cdot> x \<cdot> a y = a (x \<cdot> y) \<cdot> x"
simps_83: fixes x7 :: "rat"   and x5 :: "msg" shows "Pri_AgrK x7 \<noteq> Hash x5"
diff_right_commute: fixes a :: "'k"   and c :: "'k"   and b :: "'k" shows "a - c - b = a - b - c"
is_unifiers_def: fixes \<sigma> :: "'s"   and AAA :: "'a set set" shows "is_unifiers \<sigma> AAA = (\<forall>AA\<in>AAA. is_unifier \<sigma> AA)"
stable_translation_length_nonneg: fixes f :: "'b \<Rightarrow> 'b" shows "0 \<le> stable_translation_length f"
set_4: fixes x4 :: "'a" shows "RID\<^sub>V (Rid x4) = {x4}"
row_carrier_vec: fixes i :: "nat"   and nr :: "nat"   and A :: "'b mat"   and nc :: "nat" assumes "i < nr"   and "A \<in> carrier_mat nr nc" shows "row A i \<in> carrier_vec nc"
I: fixes x :: "'a"   and y :: "'a" assumes "x \<in> {a..b}"   and "y \<in> {a..b}" shows "(f x = f y) = (x = y)"
trans: fixes b :: "'a Word.word"   and a :: "'a Word.word"   and c :: "'a Word.word" assumes "b \<le>s a"   and "c \<le>s b" shows "c \<le>s a"
rreq_rrep_sn_simps_1: fixes hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool" shows "rreq_rrep_sn (Rreq hops dip dsn dsk oip osn sip handled) = (1 \<le> osn)"
saturated_imp_saturated_RP: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "FL.empty_ord.saturated (Liminf_llist (lmap lclss_of_state Sts))"   and "FL.passive_subset (Liminf_llist (lmap lclss_of_state Sts)) = {}" shows "src.saturated_upto Sts (\<G>_Fset (N_of_state (Liminf_state Sts) \<union> P_of_state (Liminf_state Sts) \<union> Q_of_state (Liminf_state Sts)))"
max_enat_simps_5: fixes q :: "enat" shows "max \<infinity> q = \<infinity>"
simps_2: fixes a :: "('a, 'b) Combinators" shows "member a DenyAll = (a = DenyAll)"
actPres: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and \<alpha> :: "act" assumes "(P, Q) \<in> Rel" shows "\<alpha>.P \<leadsto>\<^sup>^<Rel> \<alpha>.Q"
connected_punctured_convex: fixes S :: "'a set"   and a :: "'a" assumes "convex S"   and "aff_dim S \<noteq> 1" shows "connected (S - {a})"
lens_indep_left_comp: fixes z :: "'e \<Longrightarrow> 'f"   and x :: "'g \<Longrightarrow> 'e"   and y :: "'h \<Longrightarrow> 'e" assumes "mwb_lens z"   and "x \<bowtie> y" shows "x ;\<^sub>L z \<bowtie> y ;\<^sub>L z"
card_0_eq: fixes A :: "'a set" assumes "finite A" shows "(card A = 0) = (A = {})"
exhaust: fixes y :: "('e, 'f) labeled_graph" obtains x1 :: "('e \<times> 'f \<times> 'f) set" and x2 :: "'f set" where "y = LG x1 x2"
H_bound: shows "\<forall>m\<in>#hmsetmset Ma. local.decode 0 m < local.decode 0 H"
o_bij: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'e" assumes "g \<circ> f = id"   and "f \<circ> g = id" shows "bij f"
list_all_length: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "list_all P xs = (\<forall>n<length xs. P (xs ! n))"
m_rcancel: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int" assumes "a \<noteq> \<zero>"   and "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "c \<in> carrier Zp" shows "(b \<otimes> a = c \<otimes> a) = (b = c)"
ord_spmf_map_spmfI: fixes p :: "'a spmf"   and q :: "'a spmf"   and f :: "'a \<Rightarrow> 'b" assumes "ord_spmf (=) p q" shows "ord_spmf (=) (map_spmf f p) (map_spmf f q)"
induced_map_fmpt: fixes A :: "'a set" assumes "A \<in> sets MA" shows "fmpt (restrict_space MA A) (A.induced_map A)"
open_Compl: fixes S :: "'c set" assumes "closed S" shows "open (- S)"
maxsimpI: fixes x :: "'a set" assumes "x \<in> X"   and "\<And>z. \<lbrakk>z \<in> X; order.greater_eq z x\<rbrakk> \<Longrightarrow> z = x" shows "domain.chamber x"
c_hd_aux3: fixes u :: "nat" assumes "0 < u" shows "c_hd u = (if c_len u = 1 then c_snd (u - 1) else c_fst (c_snd (u - 1)))"
seq_plus_simp: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and t :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "t \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "(s \<oplus>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> t) k = s k \<oplus> t k"
prime_is_sum4sq: fixes p :: "nat" assumes "prime p" shows "is_sum4sq_nat p"
Ball_Set: fixes t :: "('b, unit) RBT.rbt"   and P :: "'b \<Rightarrow> bool" shows "Ball (Set t) P = RBT.foldi (\<lambda>s. s = True) (\<lambda>k v s. s \<and> P k) t True"
mul_cn: fixes P :: "pol"   and Q :: "pol" assumes "isnorm P"   and "isnorm Q" shows "isnorm (P \<langle>*\<rangle> Q)"
qAbs_alphaAbs_iff: fixes xs :: "'h"   and x :: "'i"   and X :: "('f, 'g, 'h, 'i, 'j) qTerm"   and A' :: "('f, 'g, 'h, 'i, 'j) qAbs" assumes "FixVars TYPE('d) TYPE('e)" shows "(qAbs xs x X $= A') = (\<exists>x' y X'. A' = qAbs xs x' X' \<and> y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
has_bochner_integral_minus: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. - f x) (- x)"
map_poly_gcd: fixes p :: "'a poly"   and q :: "'a poly" shows "map_poly hom (gcd p q) = gcd (map_poly hom p) (map_poly hom q)"
continuous_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'd" shows "continuous F f = (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
f_Exec_Stream_Acc_Output_length: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" assumes "0 < k" shows "length (f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c) = length xs"
statEqTransition: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<longmapsto> Rs"   and "\<Psi> \<simeq> \<Psi>'" shows "\<Psi>' \<rhd> P \<longmapsto> Rs"
pivot_positions_gen_def: fixes zer :: "'b"   and A :: "'b mat" shows "pivot_positions_gen zer A \<equiv> pivot_positions_main_gen zer A (dim_row A) (dim_col A) 0 0"
mprefix_Par_Int_skip: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and B :: "'a set" shows "(Mprefix A P \<lbrakk>B\<rbrakk> SKIP) = Mprefix (A - B) (\<lambda>x. P x \<lbrakk>B\<rbrakk> SKIP)"
map_scale_distrib_right: fixes k :: "'c"   and l :: "'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "(k + l) \<cdot> t = k \<cdot> t + l \<cdot> t"
conventional_inv: fixes n :: "'node"   and ns :: "'node list"   and m :: "'node"   and v :: "'val"   and x :: "'node"   and v' :: "'val" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n \<notin> set (tl ns)"   and "v \<in> step.allDefs g n"   and "v \<in> step.allUses g m"   and "x \<in> set (tl ns)"   and "v' \<in> step.allDefs g x" shows "var g v' \<noteq> var g v"
assms_3: fixes R :: "ccs"   and S :: "ccs" assumes "(R, S) \<in> X" shows "(S, R) \<in> X"
measurable_merge: fixes I :: "'c set"   and J :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "merge I J \<in> Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M \<rightarrow>\<^sub>M Pi\<^sub>M (I \<union> J) M"
step: fixes b :: "'a"   and a :: "'a" assumes "(0::'a) < b" shows "P (Fract a b)"
abs_homogenous: fixes x :: "'a"   and a :: "real" assumes "x \<in> V" shows "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>"
partial_deriv_integrable: shows "partial_vector_derivative (\<lambda>a. F a \<bullet> j) i integrable_on Dx_pair"
carrier_coeff_def: fixes R :: "('d, 'e) ring_scheme"   and P :: "'f multiset \<Rightarrow> 'd" assumes "ring R" shows "ring.carrier_coeff R P = (\<forall>m. P m \<in> carrier R)"
generated_ringI: fixes C :: "'a set set"   and a :: "'a set" assumes "finite C"   and "disjoint C"   and "C \<subseteq> M"   and "a = \<Union> C" shows "a \<in> generated_ring"
normalize_coprime: fixes r :: "int \<times> int"   and p :: "int"   and q :: "int" assumes "Rat.normalize r = (p, q)" shows "algebraic_semidom_class.coprime p q"
intro: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd" assumes "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"   and "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"   and "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F" shows "identity_pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F"
"2_1": shows "v \<le>\<^sub>F\<^sub>I w"
fls_lr_inverse_subdegree_1: fixes x :: "'a"   and f :: "'a fls" assumes "x \<noteq> (0::'a)" shows "fls_subdegree (fls_left_inverse f x) = - fls_subdegree f"
hyps_3: fixes b :: "trm" assumes "t \<Down> b" shows "a = b"
mat_of_rows_index: fixes i :: "nat"   and rs :: "'b vec list"   and j :: "nat"   and n :: "nat" assumes "i < length rs"   and "j < n" shows "mat_of_rows n rs $$ (i, j) = rs ! i $ j"
tarjan_params_simps_7: fixes G :: "('c, 'd) graph_rec_scheme" shows "is_break (Tarjan_def.tarjan_params G) = (\<lambda>s. False)"
wp_While_True: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "bval b s" shows "wp (WHILE b DO c) Q s = wp (c;; WHILE b DO c) Q s"
PossibleWorld_def: fixes x :: "\<kappa>" shows "PossibleWorld x \<equiv> Situation x \<^bold>& \<^bold>\<diamond>(\<^bold>\<forall>p. x \<^bold>\<Sigma> p \<^bold>\<equiv> p)"
slp_indexl_lookup_def: fixes vsl :: "nat"   and M :: "('a, nat) mapping"   and a :: "'a" shows "slp_indexl_lookup vsl M a = slp_indexl vsl (the (Mapping.lookup M a))"
add_zmset_add_single: fixes a :: "'b"   and A :: "'b zmultiset" shows "add_zmset a A = A + {#a#}\<^sub>z"
vrange_vconverse: fixes r :: "V" shows "\<R>\<^sub>\<circ> (r\<inverse>\<^sub>\<circ>) = \<D>\<^sub>\<circ> r"
compile_correct0: fixes rs :: "(name \<times> pterm) fset"   and u :: "sterm"   and u' :: "sterm"   and C :: "(name, dt_def) fmap" assumes "Rewriting_Sterm.compile rs \<turnstile>\<^sub>s u \<longrightarrow> u'"   and "prules C rs" shows "rs \<turnstile>\<^sub>p sterm_to_pterm u \<longrightarrow> sterm_to_pterm u'"
ltake_lzip: fixes n :: "enat"   and xs :: "'c llist"   and ys :: "'d llist" shows "ltake n (lzip xs ys) = lzip (ltake n xs) (ltake n ys)"
sets_completionI_sets: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> sets M" shows "A \<in> sets (completion M)"
pt_swap: fixes a :: "'b"   and b :: "'b"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "[(a, b)] \<bullet> x = [(b, a)] \<bullet> x"
assms_2: shows "t adds\<^sub>p v \<ominus> s"
List_Ref: fixes h :: "'b \<Rightarrow> 'b ref"   and a :: "'b"   and as :: "'b list" shows "List h (Ref a) as = (\<exists>bs. as = a # bs \<and> List h (h a) bs)"
dvd_prod_mset: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and A :: "'a multiset" assumes "class.comm_monoid_mult times one"   and "x \<in># A" shows "dvd.dvd times x (comm_monoid_mult.prod_mset times one A)"
mem_val_alt_10_unchanged3: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 ((pa AND 68719476732) + 3) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 10 ((pa AND 68719476732) + 3) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
the_unique_root_eqI_1: fixes plr :: "int poly \<times> rat \<times> rat"   and y :: "real" assumes "\<exists>!x. root_cond plr x"   and "root_cond plr y" shows "y = real_of_1 plr"
simps: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and axs :: "Core.term set" shows "wf_theory (\<Sigma>, axs) = ((\<forall>p\<in>axs. wt_term \<Sigma> p \<and> \<turnstile>\<^sub>\<tau> p : propT) \<and> is_std_sig \<Sigma> \<and> wf_sig \<Sigma> \<and> eq_axs \<subseteq> axs)"
inf_chain_offset: fixes p :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "inf_chain p f" shows "inf_chain p (\<lambda>j. f (j + i))"
hurwitz_zeta_1: fixes c :: "real" shows "hurwitz_zeta c 1 = 0"
analz_image_freshK_simps_61: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
comp_assoc: fixes h :: "'j"   and g :: "'j"   and f :: "'j" shows "(h \<cdot>\<^sub>J g) \<cdot>\<^sub>J f = h \<cdot>\<^sub>J g \<cdot>\<^sub>J f"
less_irrefl: fixes x :: "char" shows "\<not> less_char x x"
sub_branch_mem: fixes ps :: "('d, 'e) fm list"   and i :: "'e"   and branch :: "(('d, 'e) fm list \<times> 'e) list"   and f :: "'e \<Rightarrow> 'f" assumes "(ps, i) \<in>. branch" shows "(sub_list f ps, f i) \<in>. sub_branch f branch"
ground_n_entails: shows "\<G>_Fset N \<Turnstile>G \<G>_F B"
pw_EASSERT_1: fixes \<Phi> :: "bool" shows "nofail (EASSERT \<Phi>) = \<Phi>"
exhaust_sel: fixes option :: "\<langle>'a\<rangle>\<^sub>\<bottom>" obtains "option = \<bottom>"   | "option = \<lfloor>the option\<rfloor>"
simps_2: fixes p :: "nat" shows "subd (Suc p) = frag_extend (\<lambda>f. simplicial_cone (Suc p) (\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j f i) / real (Suc p + 1)) (simplicial_subdivision (Suc p) (frag_of f) - frag_of f - subd p (chain_boundary (Suc p) (frag_of f))))"
PUF: shows "{p. p permutes {0..<n}} \<subseteq> {f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<n}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
intro: fixes D :: "('c, 'd) Order_scheme" assumes "Order D"   and "Torder_axioms D" shows "Torder D"
element_match: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "matrix_match A1 A2 B1 B2" shows "\<forall>i<row_length A1 * row_length B1. \<forall>j<length A2 * length B2. (A1 \<circ> A2 \<otimes> B1 \<circ> B2) ! j ! i = ((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2)) ! j ! i"