hcju/isabelleps huggingface dataset:
intros_15: fixes c1 :: "i32"   and c2 :: "i32"   and iop :: "relop_i" shows "\<lparr>[$C ConstInt32 c1, $C ConstInt32 c2, $Relop_i T_i32 iop]\<rparr> \<leadsto> \<lparr>[$C ConstInt32 (wasm_bool (app_relop_i iop c1 c2))]\<rparr>"
abs_poly_def: fixes x :: "'a poly" shows "\<bar>x\<bar> = (if x < 0 then - x else x)"
lead_coeff_nonzero: fixes p :: "'a mpoly" assumes "p \<noteq> 0" shows "Symmetric_Polynomials.lead_coeff p \<noteq> (0::'a)"
concurrentD2: fixes s1 :: "'a \<times> ('a, 'b) operation"   and s2 :: "'a \<times> ('a, 'b) operation" assumes "hb.concurrent s1 s2" shows "\<not> hb s2 s1"
sets_restrict_space_subset: fixes S :: "'a set"   and M :: "'a measure" assumes "S \<in> sets (completion M)" shows "sets (restrict_space (completion M) S) \<subseteq> sets (completion M)"
Fields_i_i_oI: fixes x :: "'a prog"   and xa :: "String.literal"   and xb :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "x \<turnstile> xa has_fields xb" shows "pred.eval (Fields_i_i_o x xa) xb"
sumAssoc: fixes P :: "ccs"   and Q :: "ccs"   and R :: "ccs" shows "P \<oplus> Q \<oplus> R \<sim> P \<oplus> (Q \<oplus> R)"
MRList2: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and c :: "((int \<times> int) set set, int) Combinators list"   and b :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C a)" shows "applied_rule_rev C x (c @ b @ [a]) = \<lfloor>a\<rfloor>"
FormP_Ex: fixes t :: "tm"   and i :: "name"   and x :: "tm" shows "{FormP t, AbstFormP \<guillemotleft>Var i\<guillemotright> Zero t x} \<turnstile> FormP (Q_Ex x)"
emeasure_restricted: fixes S :: "'c set"   and M :: "'c measure"   and X :: "'c set" assumes "S \<in> sets M"   and "X \<in> sets M" shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
map_butlast: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "map f (butlast xs) = butlast (map f xs)"
exhaust: fixes y :: "('a, 'b) sequent" obtains x1 :: "('a, 'b) form multiset" and x2 :: "('a, 'b) form multiset" where "y = ( x1 \<Rightarrow>* x2)"
relpow_E2: fixes x :: "'a"   and z :: "'a"   and n :: "nat"   and R :: "('a \<times> 'a) set" assumes "(x, z) \<in> R ^^ n" obtains "n = 0" and "x = z"   | y :: "'a" and m :: "nat" where "n = Suc m" and "(x, y) \<in> R" and "(y, z) \<in> R ^^ m"
assms: shows "LeftDerivationFix [A] 0 D (length u) (u @ [B] @ v)"
take_butlast: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "take n (butlast xs) = take n xs"
properfactor_trans2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "properfactor G a b"   and "b divides c"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "properfactor G a c"
conversep_MP_Rel_OO_eq_m: shows "MP_Rel\<inverse>\<inverse> OO (=m) = MP_Rel\<inverse>\<inverse>"
Q: shows "wp\<^sub>3 c (wp\<^sub>3 (WHILE b DO c) Q) (ps, n - 1)"
paracompact_closedin: fixes U :: "'a set"   and S :: "'a set"   and \<C> :: "'a set set" assumes "closedin (top_of_set U) S"   and "\<And>T. T \<in> \<C> \<Longrightarrow> openin (top_of_set U) T"   and "S \<subseteq> \<Union> \<C>" obtains \<C>' :: "'a set set" where "S \<subseteq> \<Union> \<C>'" and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)" and "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and> finite {X \<in> \<C>'. X \<inter> V \<noteq> {}}"
induct: fixes P :: "'b heap list \<Rightarrow> bool"   and a0 :: "'b heap list" assumes "P []"   and "\<And>uu_. P [uu_]"   and "\<And>h1 h2. P [h1, h2]"   and "\<And>h1 h2 v va. P (v # va) \<Longrightarrow> P (h1 # h2 # v # va)" shows "P a0"
adv: fixes \<A> :: "nat \<Rightarrow> ('a + 'e, 'b + 'f) distinguisher" assumes "\<And>\<eta>. \<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>g \<A> \<eta> \<surd>"   and "\<And>\<eta>. interaction_any_bounded_by (\<A> \<eta>) (bound \<eta>)"   and "\<And>\<eta>. lossless \<Longrightarrow> plossless_gpv (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) (\<A> \<eta>)" shows "Negligible.negligible (\<lambda>\<eta>. advantage (\<A> \<eta>) (sim \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal_resource \<eta>) (real_resource \<eta>))"
dw_closedD: fixes A :: "'a set"   and a :: "'a"   and b :: "'a" assumes "dw_closed A"   and "a \<in> A"   and "a \<rightarrow> b" shows "b \<in> A"
prems_10: shows "\<alpha> = Cantor_sum \<beta>s ns"
a2n_n2a: fixes u :: "nat discr" shows "a2n\<cdot>(n2a\<cdot>u) = up\<cdot>u"
ae: shows "AE x in M. g x \<le> c * f x"
cSUP_least: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset"   and M :: "'d multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "\<Squnion> (f ` A) \<subseteq># M"
induct: fixes P :: "(nat \<times> (nat \<times> 'a) list) list \<Rightarrow> (nat \<times> (nat \<times> 'a) list) list \<Rightarrow> bool"   and a0 :: "(nat \<times> (nat \<times> 'a) list) list"   and a1 :: "(nat \<times> (nat \<times> 'a) list) list" assumes "\<And>bs. P [] bs"   and "\<And>v va. P (v # va) []"   and "\<And>i a as j b bs. \<lbrakk>i < j \<Longrightarrow> P as ((j, b) # bs); \<lbrakk>\<not> i < j; j < i\<rbrakk> \<Longrightarrow> P ((i, a) # as) bs; \<lbrakk>\<not> i < j; \<not> j < i\<rbrakk> \<Longrightarrow> P as bs\<rbrakk> \<Longrightarrow> P ((i, a) # as) ((j, b) # bs)" shows "P a0 a1"
mk_safe_coll_def: fixes X :: "'c set" shows "mk_safe_coll X = sets_of_coll X \<bind> (\<lambda>XS. FORWEAK XS (RETURN op_empty_coll) (\<lambda>x. mk_safe x \<bind> (\<lambda>s. RETURN (mk_coll s))) (\<lambda>b c. RETURN (b \<union> c)))"
mc_unit: fixes x :: "mc"   and y :: "mc" shows "x = y"
Pg_le_def: fixes G :: "Pg"   and H :: "Pg" shows "(G \<le> H) = (\<exists>g h. g \<in> Rep_Pg G \<and> h \<in> Rep_Pg H \<and> ge_game (h, g))"
dvdE: fixes times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and b :: "'g"   and a :: "'g" assumes "dvd.dvd times b a" obtains k :: "'g" where "a = times b k"
c: shows "(INF n\<in>{1..}. Sup (u ` {n..})) = (INF n. Sup (u ` {n..}))"
E_extendsR: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} c { F}" shows "\<turnstile>\<^sub>3\<^sub>' {P} c { F}"
F'_shift: shows "derivation_shift F' (Suc j) 0 = take (n - m) F"
NetsCollectedSep: fixes p :: "('b, 'c) Combinators list" assumes "noDenyAll1 p"   and "NetsCollected p" shows "NetsCollected (separate p)"
solution_uminus_right: fixes x :: "'a"   and y :: "'a" shows "solution (x, - y) = solution (x, y)"
nat_cut_le_finite: fixes I :: "nat set"   and t :: "nat" shows "finite (I \<down>\<le> t)"
nsum_ideal_excTr: fixes A :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A" shows "(\<forall>i\<le>n. f i \<in> carrier R) \<and> (\<exists>j\<le>n. (\<forall>l\<in>{i. i \<le> n} - {j}. f l \<in> A) \<and> f j \<notin> A) \<longrightarrow> \<Sigma>\<^sub>e R f n \<notin> A"
hom_0_iff: fixes x :: "rat" shows "(of_rat x = (0::'a)) = (x = 0)"
subst_type_term_without_type_insts_eq_subst_term: fixes insts :: "((variable \<times> typ) \<times> Core.term) list"   and t :: "Core.term" shows "subst_type_term [] insts t = subst_term insts t"
digraph_isomorphism_invI: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "pre_digraph.digraph_isomorphism (app_iso hom G) (inv_iso hom)"
equiv_up_to_weaken: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "P \<Turnstile> c \<sim> c'"   and "\<And>s. P' s \<Longrightarrow> P s" shows "P' \<Turnstile> c \<sim> c'"
assms_2: shows "\<Union>\<^sub>\<circ> A \<in>\<^sub>\<circ> Vset \<alpha>"
swap_msgs_Send_Recv: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isRecv ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
bot_neq_top_formula: shows "\<bottom> \<noteq> \<top>"
i_aggregate_i_append_mod: fixes xs :: "'b list"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b"   and ag :: "'b list \<Rightarrow> 'b" assumes "length xs mod k = 0" shows "i_aggregate (xs \<frown> f) k ag = f_aggregate xs k ag \<frown> i_aggregate f k ag"
assms_5: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq s t"
2: fixes f :: "'a poly" shows "f \<noteq> 0 \<longrightarrow> degree f \<le> p \<longrightarrow> map (nth_default (0::'a) (coeffs f)) [0..<p + 1] = coeffs f @ replicate (p - degree f) (0::'a)"
f: fixes i :: "nat" assumes "F i \<noteq> {}" shows "f i \<in> F i"
bl_xor_bin: fixes n :: "nat"   and v :: "int"   and w :: "int" shows "map2 (\<noteq>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v XOR w)"
expand: fixes react :: "'c react"   and react' :: "'c react" assumes "un_React react = un_React react'" shows "react = react'"
perm_2: fixes pi :: "(name \<times> name) list"   and x2 :: "name"   and x1 :: "name" shows "pi \<bullet> x2[x1] = (pi \<bullet> x2)[(pi \<bullet> x1)]"
exec_n_Nil: fixes c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and k :: "nat"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" shows "[] \<turnstile> c \<rightarrow>^k c' = (c' = c \<and> k = 0)"
eventually_frequently_const_simps_6: fixes C :: "bool"   and P :: "'c \<Rightarrow> bool"   and F :: "'c filter" shows "(\<forall>\<^sub>F x in F. C \<longrightarrow> P x) = (C \<longrightarrow> eventually P F)"
conts_1: fixes X :: "real \<Rightarrow> 'a" assumes " D X = (\<lambda>\<tau>. f \<tau> (X \<tau>)) on U (X t\<^sub>0)" shows "continuous_on (\<P> X (U (X t\<^sub>0))) \<nu>"
pfa_append_grow_less_rule: fixes n :: "nat"   and c :: "nat"   and l :: "'b list"   and a :: "'b array"   and x :: "'b" assumes "n < c" shows "<is_pfa c l (a, n)> Array.upd n x a <\<lambda>r. is_pfa c (l @ [x]) (r, n + 1)>\<^sub>t"
map_simp: fixes f :: "'j" assumes "J.arr f" shows "\<chi>'.A.map f = x'"
circ_one: shows "mone\<^sup>\<odot> = mone\<^sup>\<odot>\<^sup>\<odot>"
vfinite_def: fixes A :: "V" shows "vfinite A = (\<exists>n\<in>elts \<omega>. n \<approx>\<^sub>\<circ> A)"
IH: fixes ka :: "nat"   and k'a :: "nat"   and v :: "'var" assumes "ka + k'a < k + k'"   and "cs\<^bsup>path \<sigma>\<^esup> ka = cs\<^bsup>path \<sigma>'\<^esup> k'a"   and "v \<in> local.reads (path \<sigma> ka)"   and "(\<sigma>\<^bsup>ka\<^esup>) v \<noteq> (\<sigma>'\<^bsup>k'a\<^esup>) v" shows "((\<sigma>, ka), \<sigma>', k'a) \<in> cp"
compl_inf: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (inf x y) = sup (uminus x) (uminus y)"
elem_eq_wrt_P: shows "Y - {elem} \<in> (\<lambda>X. X - {elem}) ` P"
le_add_same_cancel1: fixes a :: "'y multiset"   and b :: "'y multiset" shows "(a \<subseteq># a + b) = ({#} \<subseteq># b)"
rref_Gauss_Jordan: fixes A :: "(('a, 'b) vec, 'c) vec" shows "reduced_row_echelon_form (Gauss_Jordan A)"
open_box: fixes a :: "'c"   and b :: "'c" shows "open (box a b)"
addition_correct: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and lm :: "nat list" assumes "m \<noteq> n"   and "max m n < p"   and "p < length lm"   and "lm ! p = 0" shows "{\<lambda>a. a = lm} addition m n p {\<lambda>nl. addition_inv (7, nl) m n p lm}"
real_roots_of_rat_poly_def: fixes p :: "rat poly" shows "real_roots_of_rat_poly p = map real_of (roots_of_real_alg (snd (rat_to_int_poly p)))"
neg_le_0_iff_le: fixes a :: "'b" shows "(- a \<le> (0::'b)) = ((0::'b) \<le> a)"
t_eq_imp: fixes k :: "'key"   and obs' :: "'observation list"   and obs :: "'observation list" assumes "k \<in> keys"   and "K k \<noteq> 0"   and "obs' \<in> OB ` msgs"   and "obs \<in> OB ` msgs"   and "t obs = t obs'" shows "prob ((\<lambda>x. (OB x, fst x)) -` {(obs, k)} \<inter> msgs) = prob ((\<lambda>x. (OB x, fst x)) -` {(obs', k)} \<inter> msgs)"
suma2__conga: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p"   and G' :: "'p"   and H' :: "'p"   and I' :: "'p" assumes "A B C D E F SumA G H I"   and "A B C D E F SumA G' H' I'" shows "G H I CongA G' H' I'"
implicit_vector_space_ow: fixes U\<^sub>V\<^sub>S :: "'e set" shows "vector_space_ow U\<^sub>V\<^sub>S (+) (0::'e) (-) uminus = vector_space_on U\<^sub>V\<^sub>S"
arcs_H: shows "arcs H = {uw, rev_H uw, vw, rev_H vw} \<union> arcs G - {uv, rev_G uv}"
simps_2: fixes \<gamma> :: "'c \<Rightarrow> 'b set"   and a :: "'c" shows "\<gamma>_option \<gamma> (Some a) = \<gamma> a"
the_index_bounded: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "the_index xs x < length xs"
ctermsl_subterms: fixes p :: "('a, 'b, 'c, 'd) seqp" shows "ctermsl p = {q \<in> subterms p. not_choice q}"
finite_INT: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "\<exists>x\<in>I. finite (A x)" shows "finite (\<Inter> (A ` I))"
assms: shows "dt (Suc t) \<noteq> [] \<or> dt (Suc (Suc t)) \<noteq> []"
bin_sign_simps: fixes w :: "int"   and b :: "bool" shows "bin_sign (w BIT b) = bin_sign w"
bin_sbin_eq_iff: fixes n :: "nat"   and x :: "int"   and y :: "int" shows "(take_bit (Suc n) x = take_bit (Suc n) y) = (signed_take_bit n x = signed_take_bit n y)"
minimal_unsat_core_ns_def: fixes I :: "'a set"   and cs :: "('a \<times> 'b ns_constraint) set" shows "minimal_unsat_core_ns I cs = (I \<subseteq> fst ` cs \<and> (\<nexists>v. (I, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  cs) \<and> (distinct_indices_ns cs \<longrightarrow> (\<forall>J\<subset>I. \<exists>v. (J, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  cs)))"
F_def: shows "F \<equiv> \<lambda>g z. if z \<in> f ` Field s then g (the_inv_into (Field s) f z) else if z \<in> Field t then r.zero else undefined"
mk_deriv_intro_def: fixes ix :: "nat"   and n :: "nat"   and j :: "nat" shows "mk_deriv_intro ix n j = (n, j, ix)"
left_def: fixes t :: "'a" shows "left t = (case unapp t of Some (l, x) \<Rightarrow> l)"
L1_Norm_1: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "Norm (\<LL> 1 M) f = LINT x|M. \<bar>f x\<bar>"
unit_divides: fixes u :: "'a"   and a :: "'a" assumes "u \<in> Units (add_monoid R)"   and "a \<in> carrier R" shows "u divides\<^bsub>add_monoid R\<^esub> a"
Gcd_group_closure_in_group_closure: fixes S :: "int set" shows "Gcd (group_closure S) \<in> group_closure S"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>x. P x 0"   and "\<And>x i. P (x div base) i \<Longrightarrow> P x (Suc i)" shows "P a0 a1"
prefix_length_less_length: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. \<not> P x" shows "prefix_length P xs < length xs"
grgr: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) v = ([y, x], rev v @ h)"
add_edge_valid: fixes g :: "('c, 'd) graph"   and v :: "'c"   and e :: "'d"   and v' :: "'c" assumes "valid_graph g" shows "valid_graph (add_edge v e v' g)"
sub: fixes a :: "'a" assumes "a \<in> Ker" shows "a \<in> G"
funpow_simps_right_2: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'c" shows "f ^^ Suc n = f ^^ n \<circ> f"
c': shows "c = k *\<^sub>R (b - a) + a"
elims: fixes x :: "bool list"   and xa :: "trie"   and y :: "trie" assumes "prefix_trie x xa = y" obtains t :: "trie" where "x = []" and "xa = t__" and "y = t__"   | k :: "bool" and ks :: "bool list" and t :: "trie" where "x = k # ks" and "xa = t" and "y = (let t' = prefix_trie ks t in Nd False (if k then (Lf, t') else (t', Lf)))"
f1''_def: shows "f1'' = f1' * real l0 / real (l0 + 1)"
assms_2: shows "h \<turnstile> create_document \<rightarrow>\<^sub>h h'"
equality: fixes r :: "('a, 'b) state_scheme"   and r' :: "('a, 'b) state_scheme" assumes "globals r = globals r'"   and "state.more r = state.more r'" shows "r = r'"
real_integrable_def: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "integrable M f = (f \<in> borel_measurable M \<and> \<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<noteq> \<infinity> \<and> \<integral>\<^sup>+ x. ennreal (- f x) \<partial>M \<noteq> \<infinity>)"
cupcake_build_rec_preserve: fixes cl_env :: "v sem_env"   and env :: "(char list, char list, v) namespace"   and fs :: "(char list \<times> char list \<times> exp0) list" assumes "is_cupcake_all_env cl_env"   and "is_cupcake_ns env"   and "list_all (\<lambda>(uu_, uu_, e). is_cupcake_exp e) fs" shows "is_cupcake_ns (build_rec_env fs cl_env env)"
Finite_Graph_EI: assumes "finite cf.E" shows "Finite_Graph cf"
power_minus_even: fixes a :: "'a"   and n :: "nat" shows "(- a) ^ (2 * n) = a ^ (2 * n)"
gVarGOpInj_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gVarGOpInj MOD \<equiv> \<forall>xs x delta inp' binp' inp binp. wlsInp delta inp' \<and> wlsBinp delta binp' \<and> gWlsInp MOD delta inp \<and> gWlsBinp MOD delta binp \<and> asSort xs = stOf delta \<longrightarrow> gVar MOD xs x \<noteq> gOp MOD delta inp' inp binp' binp"
yields_isomorphic_representation': obtains \<psi> :: "'a" where "\<guillemotleft>\<psi> : g \<star> f\<^sup>* \<Rightarrow> g \<star> f\<^sup>*\<guillemotright>" and "local.iso \<psi>"
1: shows "y \<squnion> y\<^sup>T = x \<and> pd_kleene_allegory_class.acyclic y \<and> injective y \<and> univalent y"
renaming_is_injective: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'a set"   and L1 :: "'a Literal"   and L2 :: "'a Literal" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.rename_literal A L1 = propositional_atoms.rename_literal A L2" shows "L1 = L2"
mono_transD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "mono_trans t"   and "sound P"   and "sound Q"   and "P \<tturnstile> Q" shows "t P \<tturnstile> t Q"
trivial_morphism: fixes X :: "'b set set" assumes "ChamberComplex X" shows "ChamberComplexMorphism X X id"
converge1: shows "cs\<^bsup>\<pi>\<^sub>1\<^esup> n = cs\<^bsup>\<pi>\<^sub>1'\<^esup> n'"
ltl_prop_equiv_implies_ltl_prop_unfold_equiv: fixes \<phi> :: "'d ltln"   and \<psi> :: "'d ltln" assumes "\<phi> \<sim>\<^sub>P \<psi>" shows "\<phi> \<sim>\<^sub>Q \<psi>"
inverse_arrows_sym: fixes f :: "'b"   and g :: "'b" shows "B.inverse_arrows f g = B.inverse_arrows g f"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P merge_guards c Q,A"
lexordp_conv_lexordp_eq: fixes xs :: "'d word list"   and ys :: "'d word list" shows "signed.lexordp xs ys = Restricted_Predicates.strict signed.lexordp_eq xs ys"
M_inv_M_id: fixes m :: "int"   and x :: "int" shows "poly_mod.M m (poly_mod.inv_M m x) = poly_mod.M m x"
mcont1: shows "mcont (flat_lub None) option_ord lub ord f"
one_dim_iso_add: fixes a :: "'c"   and b :: "'c" shows "one_dim_iso (a + b) = one_dim_iso a + one_dim_iso b"
row_def: fixes A :: "'b mat"   and i :: "nat" shows "Matrix.row A i = Matrix.vec (dim_col A) (\<lambda>j. A $$ (i, j))"
simps_4: fixes bs :: "real list"   and a :: "MIR.num" shows "Inum bs (Neg a) = - Inum bs a"
g_def: shows "g = r.worec (\<lambda>g a. s.suc (g ` r.underS a))"
respects_unit: fixes a :: "'b" assumes "B.obj a" shows "(\<tau>\<^sub>0 a \<star>\<^sub>C \<tau>.F.unit a) \<cdot>\<^sub>C \<r>\<^sub>C\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>C \<l>\<^sub>C[\<tau>\<^sub>0 a] = \<tau>\<^sub>1 a \<cdot>\<^sub>C (\<tau>.G.unit a \<star>\<^sub>C \<tau>\<^sub>0 a)"
zip: shows "zip (bia @ [a]) [0..<length (bia @ [a])] = zip bia [0..<length bia] @ [(a, length bia)]"
exhaust: fixes y :: "'b top" obtains x1 :: "'b" where "y = top.Value x1"   | "y = top.Top"
lt_refl_Integer: fixes x :: "Integer" assumes "x \<noteq> \<bottom>" shows "lt\<cdot>x\<cdot>x = FF"
max_wp_repeat: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "maximal (wp a)" shows "maximal (wp (repeat n a))"
empty_edges: shows "sinvar_spec \<lparr>nodes = set (nodesL G), edges = {}\<rparr> nP"
assms: fixes i :: "nat" assumes "i < max_Var_floatarith f" shows "xs ! i = ys ! i"
add_mono_thms_linordered_semiring_2: fixes i :: "'d"   and j :: "'d"   and k :: "'d"   and l :: "'d" assumes "i = j \<and> k \<le> l" shows "i + k \<le> j + l"
simps_2: fixes A :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "mk_mon A (Suc n) = A (Suc n) \<union> mk_mon A n"
deg_nzero_nzero: fixes p :: "nat \<Rightarrow> int" assumes "deg R p \<noteq> 0" shows "p \<noteq> \<zero>\<^bsub>UP R\<^esub>"
\<Delta>'''_sel_2: fixes x :: "'v"   and y :: "'v" shows "capacity \<Delta>''' (x, y) = (if x = sink \<Delta> \<or> y = source \<Delta> then 0 else capacity \<Delta> (x, y))"
inverse_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse one = one"
inspan: fixes h :: "'a" shows "P h \<in> cspan {t}"
simps: fixes ant :: "'a form multiset"   and suc :: "'a form" shows "antec ( ant \<Rightarrow>* suc) = ant"
termi_c: shows "\<Gamma>\<turnstile>c \<down> s"
one_zero_def: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "one_zero A i j \<equiv> (Suc i < j \<longrightarrow> A $$ (i, Suc i) = (1::'b) \<longrightarrow> A $$ (i, j) = (0::'b)) \<and> (j < i \<longrightarrow> A $$ (i, j) = (0::'b)) \<and> (i < j \<longrightarrow> A $$ (i, i) \<noteq> A $$ (j, j) \<longrightarrow> A $$ (i, j) = (0::'b))"
attach_stateless_callee: fixes callee :: "'c \<Rightarrow> ('d, 'e, 'f) gpv"   and "oracle" :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b) spmf" shows "attach_callee (stateless_callee callee) oracle = \<dagger>(\<lambda>s q. exec_gpv oracle (callee q) s)"
is_zero_def: fixes p :: "'c \<Rightarrow>\<^sub>0 'd" shows "is_zero p = (p = 0)"
Smith_normal_form_upt_k_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" shows "Smith_normal_form_upt_k A k = ((\<forall>a b. mod_type_class.to_nat a = mod_type_class.to_nat b \<and> mod_type_class.to_nat a + 1 < k \<and> mod_type_class.to_nat b + 1 < k \<longrightarrow> A $ a $ b dvd A $ (a + (1::'f)) $ (b + (1::'e))) \<and> isDiagonal_upt_k A k)"
l_inv: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
M'_1: fixes l :: "nat" shows "M' l \<subseteq> closure (V l)"
white_def: fixes r :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" shows "white r s \<equiv> obj_at (\<lambda>obj. obj_mark obj \<noteq> sys_fM s) r s"
IdNatTransMapR: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans f" shows "f \<bullet> IdNatTrans (NTCod f) = f"
entails_trans: fixes N1 :: "'a set"   and N2 :: "'a set"   and N3 :: "'a set" assumes "N1 \<Turnstile> N2"   and "N2 \<Turnstile> N3" shows "N1 \<Turnstile> N3"
hyps_2: shows "\<not> AlternativeSmallStep.inf \<Gamma> [c1] [] (Normal s)"
CDMode_CTRL_def: shows "CDMode_CTRL = Abs_seqauto (CDMode_CTRL_States, CDMode_CTRL_Init, CDMode_CTRL_Labels, CDMode_CTRL_Delta)"
r_into_rtrancl: fixes p :: "'d \<times> 'd"   and r :: "('d \<times> 'd) set" assumes "p \<in> r" shows "p \<in> r\<^sup>*"