hcju/isabelleps huggingface dataset:
subclsR_code: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P \<turnstile> C \<prec>\<^sub>R D = subclsRp P C D"
z_lemma_R: fixes sign :: "rat list"   and p :: "real poly"   and qs :: "real poly list"   and I :: "nat list \<times> nat list" assumes "sign \<in> set (characterize_consistent_signs_at_roots p qs)"   and "list_constr (fst I) (length qs)"   and "list_constr (snd I) (length qs)" shows "z_R I sign = 1 \<or> z_R I sign = 0 \<or> z_R I sign = - 1"
next_state_consistent: shows "\<forall>s t u execs. vpeq (current s) s t \<and> vpeq u s t \<and> current s = current t \<longrightarrow> vpeq u (next_state s execs) (next_state t execs)"
eval_gext': fixes G :: "prog"   and x1 :: "abopt"   and s1 :: "st"   and t :: "Term.term"   and w :: "vals"   and x2 :: "abopt"   and s2 :: "st" assumes "G\<turnstile>(x1, s1) \<midarrow>t\<succ>\<rightarrow> (w, (x2, s2))" shows "s1\<le>|s2"
ide_dom: fixes f :: "'s" assumes "arr f" shows "ide (local.dom f)"
cases: fixes x :: "'d net_state" obtains i :: "nat" and s :: "'d" and R :: "nat set" where "x = NodeS i s R"   | n1 :: "'d net_state" and n2 :: "'d net_state" where "x = SubnetS n1 n2"
merges_preserves_type_lemma: fixes ps :: "(nat \<times> nat list) list" shows "\<forall>xs. xs \<in> list n A \<longrightarrow> (\<forall>(p, x)\<in>set ps. p < n \<and> x \<in> A) \<longrightarrow> merges f ps xs \<in> list n A"
mult_vv_scale_sv2: fixes v1 :: "complex \<times> complex"   and k :: "complex"   and v2 :: "complex \<times> complex" shows "v1 *\<^sub>v\<^sub>v (k *\<^sub>s\<^sub>v v2) = k * v1 *\<^sub>v\<^sub>v v2"
nxtAct_active: fixes c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat" assumes "\<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "\<langle>c \<rightarrow> t\<rangle>\<^bsub>i\<^esub> = i"
different_images: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and b :: "'d" assumes "f a \<noteq> f b" shows "a \<noteq> b"
subresultant_product: fixes F :: "'b poly"   and B :: "'b poly"   and G :: "'b poly"   and J :: "nat" assumes "F = B * G"   and "degree G \<le> degree F" shows "subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)"
iso_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C" shows "category.iso C f \<equiv> \<exists>g. category.inverse_arrows C f g"
connected_component_sym: fixes S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "connected_component S x y" shows "connected_component S y x"
simps_2: fixes n :: "nat"   and ns :: "nat list"   and f :: "face"   and fs :: "face list"   and F :: "face list list" shows "replacefacesAt2 (n # ns) f fs F = (if n < |F| then replacefacesAt2 ns f fs (F[n := replace f fs (F ! n)]) else replacefacesAt2 ns f fs F)"
f_def: shows "f = (\<lambda>g t. Im (g t - z) / Re (g t - z))"
f_def: fixes x :: "'b" shows "f x = (THE y. R y x)"
ProblemIk_axioms: shows "ProblemIk GE_wtFsym GE_wtPsym GE_arOf GE_resOf GE_parOf gPB infTp"
from_hma_to_hma\<^sub>m: fixes A :: "'a mat" assumes "A \<in> carrier_mat CARD('b) CARD('c)" shows "from_hma\<^sub>m (to_hma\<^sub>m A) = A"
comp_transpos: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "\<forall>k\<le>n. compose {i. i \<le> n} (transpos i j) (transpos i j) k = k"
CollectI: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "P a" shows "a \<in> \<lbrace>\<acute>P\<rbrace>"
match: shows "mC_ZOC ZObis c1 d1 s t I0 P F"
param_optci_def: fixes i :: "nat"   and t :: "nat"   and e :: "nat"   and bdd :: "bddi" shows "param_optci i t e bdd = tci bdd \<bind> (\<lambda>(tr, bdd). fci bdd \<bind> (\<lambda>(fl, bdd). destrci i bdd \<bind> (\<lambda>id. destrci t bdd \<bind> (\<lambda>td. destrci e bdd \<bind> (\<lambda>ed. return (if id = TD then Some t else if id = FD then Some e else if td = TD \<and> ed = FD then Some i else if t = e then Some t else if ed = TD \<and> i = t then Some tr else if td = FD \<and> i = e then Some fl else None, bdd))))))"
strict_mono_imp_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "strict_mono f" shows "inj_on f A"
indep_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E>.indep_in \<E> X" shows "X \<subseteq> \<E>"
IH: assumes "list_constr xa (length qs)" shows "(prod_list (map ((!) sign) xa) = 0) = (poly (prod_list (retrieve_polys qs xa)) x = 0)"
H_2: fixes U :: "'a set" assumes "U \<in> sets M"   and "emeasure M U < ennreal epsilon" shows "\<bar>set_lebesgue_integral M U f\<bar> < delta"
f'_def: fixes x :: "nat" shows "f' x = (if x \<in> to_nat_on A ` A then enn2real (f (from_nat_into A x)) else 0)"
scale_zero_right: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'d" assumes "module scale" shows "scale a (0::'e) = (0::'e)"
qty_single_updated: fixes Mkt :: "('c, 'd) discrete_market"   and asset :: "'d"   and qty :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and n :: "nat"   and w :: "'c" shows "cls_val_process Mkt (qty_single asset qty) (Suc n) w = prices Mkt asset (Suc n) w * qty (Suc n) w"
igOpIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igOpIPresIGWls MOD \<equiv> \<forall>delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igWls MOD (stOf delta) (igOp MOD delta inp binp)"
entails_ex: fixes P :: "'c \<Rightarrow> assn"   and Q :: "assn" shows "(\<not> (\<exists>\<^sub>Ax. P x \<Longrightarrow>\<^sub>A Q)) = (\<exists>x. \<not> (P x \<Longrightarrow>\<^sub>A Q))"
289: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (- x \<sqinter> y) = top"
asTerm_qSwap_swap: fixes qX :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "qGood qX" shows "asTerm (qX #[[x1 \<and> x2]]_xs) = asTerm qX #[x1 \<and> x2]_xs"
step: fixes Sup :: "'g set \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and f :: "'g \<Rightarrow> 'g" assumes "class.ccpo Sup less_eq less"   and "x \<in> ccpo.iterates Sup less_eq f" shows "f x \<in> ccpo.iterates Sup less_eq f"
freshTransition_3: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>a<b> \<prec> P'"   and "c \<sharp> P"   and "c \<noteq> b" shows "c \<sharp> P'"
right_gpv_parametric': fixes A :: "'m \<Rightarrow> 'p \<Rightarrow> bool"   and C' :: "'n \<Rightarrow> 'q \<Rightarrow> bool"   and R' :: "'o \<Rightarrow> 'r \<Rightarrow> bool"   and C :: "'s \<Rightarrow> 'u \<Rightarrow> bool"   and R :: "'t \<Rightarrow> 'v \<Rightarrow> bool" shows "(rel_gpv'' A C' R' ===> rel_gpv'' A (rel_sum C C') (rel_sum R R')) right_gpv right_gpv"
matrix_transpose_mul: fixes A :: "(('c, 'f) vec, 'd) vec"   and B :: "(('c, 'e) vec, 'f) vec" shows "Finite_Cartesian_Product.transpose (A ** B) = Finite_Cartesian_Product.transpose B ** Finite_Cartesian_Product.transpose A"
finite_blocks: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>R" shows "finite b"
conv_radius_leI_ex: fixes R :: "ereal"   and f :: "nat \<Rightarrow> 'b" assumes "0 \<le> R"   and "\<And>r. \<lbrakk>0 < r; R < ereal r\<rbrakk> \<Longrightarrow> \<exists>z. norm z = r \<and> \<not> summable (\<lambda>n. norm (f n * z ^ n))" shows "conv_radius f \<le> R"
RPDi_Times: fixes i :: "nat" shows "RPDi i (MTimes r s) \<subseteq> {MTimes r s} \<union> MTimesL r (\<Union>j\<le>i. RPDi j s) \<union> (\<Union>j\<le>i. RPDi j r)"
assms_2: shows "(k1, k2) \<noteq> (0, 0)"
sum_list_replicate0: fixes n :: "nat" shows "sum_list (replicate n (0::'a)) = (0::'a)"
termMOD_igSubstIGVar1: shows "igSubstIGVar1 termMOD"
Le_Lt_dbm_lt_D: fixes a :: "'d"   and b :: "'d" assumes "Le a \<prec> Le b" shows "a < b"
HFail_blocksOf_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "p \<noteq> q" shows "blocksOf s' q \<subseteq> blocksOf s q"
assms_2: shows "\<not> (\<exists>w\<in>fv s. t \<sqsubseteq> \<I> w)"
wiringD_1: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and cnv :: "('i, 'j, 'k, 'l) converter"   and f :: "'i \<Rightarrow> 'k"   and g :: "'l \<Rightarrow> 'j" assumes "wiring \<I> \<I>' cnv (f, g)" shows "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C"
append_Cons_nth_middle: fixes i :: "nat"   and xs :: "'b list"   and y :: "'b"   and zs :: "'b list" assumes "i = length xs" shows "(xs @ y # zs) ! i = y"
H2_CSP1_commute2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "H2 (CSP1 P) = CSP1 (H2 P)"
filterlim_at_within_If: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and G :: "'c filter"   and x :: "'b"   and A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'c" assumes "class.topological_space open"   and "filterlim f G (topological_space.at_within open x (A \<inter> {x. P x}))"   and "filterlim g G (topological_space.at_within open x (A \<inter> {x. \<not> P x}))" shows "LIM x topological_space.at_within open x A. if P x then f x else g x :> G"
prognosis_subst_Lam: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and y :: "Vars.var"   and x :: "Vars.var"   and S :: "stack_elem list" shows "prognosis ae as (pred\<cdot>a) (\<Gamma>, e[y::=x], S) \<sqsubseteq> prognosis ae as a (\<Gamma>, Lam [y]. e, Arg x # S)"
genPrefix_length_le: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(xs, ys) \<in> genPrefix r" shows "length xs \<le> length ys"
ffold_finsert_idem: fixes z :: "'b"   and x :: "'a"   and A :: "'a fset" shows "ffold f z (finsert x A) = f x (ffold f z A)"
Rep_trace_inverse: fixes x :: "'b trace" shows "Abs_trace (Rep_trace x) = x"
add_mset_eq_singleton_iff: fixes x :: "'a"   and M :: "'a multiset"   and y :: "'a" shows "(add_mset x M = \<LM> y  \<RM>) = (M = \<Empt> \<and> x = y)"
small_Union: fixes \<A> :: "'a set set" assumes "\<A> \<subseteq> Collect small"   and "small \<A>" shows "small (\<Union> \<A>)"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad return bind"   and "monad_state_axioms bind get put" shows "monad_state return bind get put"
sel_2: fixes x1 :: "nat"   and x2 :: "'b"   and x3 :: "'b tree list" shows "tree.root (Node x1 x2 x3) = x2"
delete_point_split_blocks: fixes p :: "'a" shows "del_point_blocks p = filter_mset ((\<notin>) p) \<B> + {#bl - {p}. bl \<in># filter_mset ((\<in>) p) \<B>#}"
ii_cvec_def: shows "ii\<^sub>v = (\<i>, 1)"
extremum_unique: fixes a :: "'f multiset" shows "(a \<subseteq># {#}) = (a = {#})"
nat_pow_eq_0_iff: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat" assumes "domain R"   and "a \<in> carrier R" shows "(a [^]\<^bsub>R\<^esub> n = \<zero>\<^bsub>R\<^esub>) = (a = \<zero>\<^bsub>R\<^esub> \<and> n \<noteq> 0)"
RelStay: fixes P :: "pi"   and Q :: "pi" assumes "(P \<parallel> !P, Q) \<in> Rel'" shows "(!P, Q) \<in> Rel'"
isomorphic_categories_axioms: shows "isomorphic_categories (\<cdot>) CC.COMP"
mset_tl: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "mset (tl xs) = mset xs - {#hd xs#}"
n_def: shows "n = Max {n\<^sub>1, n\<^sub>2, k}"
local_lipschitz_compact_bicomposeE: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and I :: "'b set"   and v :: "'b \<Rightarrow> 'c"   and w :: "'b \<Rightarrow> 'c" assumes "local_lipschitz T X f"   and "\<And>x. x \<in> X \<Longrightarrow> continuous_on I (\<lambda>t. f t x)"   and "compact I"   and "I \<subseteq> T"   and "continuous_on I v"   and "continuous_on I w"   and "v ` I \<subseteq> X"   and "w ` I \<subseteq> X" obtains L :: "real" where "0 < L" and "\<And>x. x \<in> I \<Longrightarrow> dist (f x (v x)) (f x (w x)) \<le> L * dist (v x) (w x)"
oppOrdering_compare: fixes x :: "'a"   and y :: "'a" shows "oppOrdering\<cdot>(compare\<cdot>x\<cdot>y) = compare\<cdot>y\<cdot>x"
assms_1: shows "Rewriting_Pterm.compile rs \<turnstile>\<^sub>p t \<longrightarrow> t'"
inside_simple_curve_imp_closed: fixes \<gamma> :: "real \<Rightarrow> 'a"   and x :: "'a" assumes "simple_path \<gamma>"   and "x \<in> inside (path_image \<gamma>)" shows "pathfinish \<gamma> = pathstart \<gamma>"
cop_F_closed_inv_def: fixes ds :: "'d set"   and fp :: "'x set" shows "cop_F_closed_inv ds fp = (\<forall>x\<in>fp. above (Pd (Xd x)) x \<subseteq> fp)"
elims: fixes x :: "'a list"   and y :: "'a list" assumes "remdups_adj x = y" obtains "x = []" and "y = []"   | xa :: "'a" where "x = [xa__]" and "y = [xa__]"   | xa :: "'a" and ya :: "'a" and xs :: "'a list" where "x = xa # ya # xs" and "y = (if xa = ya then remdups_adj (xa # xs) else xa # remdups_adj (ya # xs))"
cases: fixes x :: "(bool, unit) arr" obtains A :: "bool" and uu_ :: "bool" and uv_ :: "unit" where "x = MkArr A uu_ uv_"   | "x = Null"
num_params_tmf_polys1: fixes z :: "float interval list" shows "num_params (fst (tmf_polys z)) \<le> Suc 0"
get_def: fixes h :: "heap" shows "Ref.get h = (from_nat \<circ>\<circ>\<circ> refs) h TYPEREP('b) \<circ> addr_of_ref"
cong_exp_iff_simps_7: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus num.One) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 q))) = (modulo (numeral.numeral one plus n) (numeral.numeral one plus q) = zero)"
c_pair_inj2: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "c_pair x1 y1 = c_pair x2 y2" shows "y1 = y2"
simps_2: fixes f :: "'b"   and T :: "('b, 'a) Term.term list" shows "fv_list (Fun f T) = concat (map fv_list T)"
fps_unit_factor_def: fixes f :: "'a fps" shows "unit_factor f = fps_shift (subdegree f) f"
add_swap_def: shows "add_swap = (\<lambda>x y. y + x)"
fmeasurable_inner_outer: fixes S :: "'a set" shows "(S \<in> fmeasurable M) = (\<forall>e>0. \<exists>T\<in>fmeasurable M. \<exists>U\<in>fmeasurable M. T \<subseteq> S \<and> S \<subseteq> U \<and> \<bar>Sigma_Algebra.measure M T - Sigma_Algebra.measure M U\<bar> < e)"
abs_eq_0_iff: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "(abs a = zero) = (a = zero)"
del_type_1: fixes t :: "'a bro"   and h :: "nat"   and x :: "'a" assumes "t \<in> B h" shows "del x t \<in> T h"
fsafe_InContext: fixes f :: "('a, 'b, 'c) formula"   and C :: "'b" assumes "fsafe f" shows "fsafe (InContext C f)"
insert_subset: fixes x :: "'e"   and A :: "'e set"   and B :: "'e set" shows "(insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
cd_is_cd_ipd: fixes k :: "nat"   and m :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and j :: "nat" assumes "k < m"   and "\<pi> m = ipd (\<pi> k)"   and "\<forall>n\<in>{k..<m}. \<pi> n \<noteq> ipd (\<pi> k)"   and "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"   and "ipd (\<pi> j) \<noteq> \<pi> m" shows "m cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
same_suffix_suffix: fixes ys :: "'c list"   and xs :: "'c list"   and zs :: "'c list" shows "suffix (ys @ xs) (zs @ xs) = suffix ys zs"
1: shows "\<guillemotleft>g' : F a \<rightarrow>\<^sub>B F a'\<guillemotright>"
hyps_1: fixes p :: "'a list" assumes "n = degree p"   and "p \<in> carrier (poly_ring R)" shows "finite (Collect (is_root p))"
F_def: shows "F = WHILEI_body (\<bind>) RETURN I cond step"
SPLIT_ACCORDING_def: fixes m :: "'c list"   and l :: "'d list" shows "SPLIT_ACCORDING m l \<equiv> length l = length m"
run_one_step_local_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x51 :: "nat"   and x52 :: "nat"   and x53 :: "v list"   and x54 :: "e list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, Local x51 x52 x53 x54) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
rq_take: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>s v" shows "v\<^sup><\<inverse>u  = take (\<^bold>|v\<^bold>| - \<^bold>|u\<^bold>|) v"
len_vertices_graph: fixes g :: "graph" shows "|vertices g| = countVertices g"
path_abc: shows "\<exists>Q\<in>\<P>. a \<in> Q \<and> b \<in> Q \<and> c \<in> Q"
prc: shows "prc (f, length ss) (g, length ts) = (False, ns)"
in_\<gamma>: shows "\<gamma> \<in> \<Gamma>"
ent_iffI: fixes A :: "assn"   and B :: "assn" assumes "A \<Longrightarrow>\<^sub>A B"   and "B \<Longrightarrow>\<^sub>A A" shows "A = B"
k': shows "X \<subseteq> Sup_upto_llist Xs (enat k')"
set: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and l :: "'b"   and v :: "('a, 'b, 'c) val" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Assign (VE (Loc l)) (VE v)])"   and "l \<in> dom (\<sigma>;;\<tau>)" shows "substitution.revision_step subst r s (s(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (CV Unit)])))"
non_det_distrib: fixes M :: "'a process"   and P :: "'a process"   and Q :: "'a process" shows "(M |-| P [+] Q) = ((M |-| P) [+] (M |-| Q))"
poly_y_x_id: fixes p :: "'a poly poly" shows "poly_y_x (poly_y_x p) = p"
bij'_def: shows "bij' = (\<lambda>a. bij (f a) a)"
A: fixes ka :: "int option"   and p :: "int \<times> 'a"   and l :: "'a tree23"   and r :: "'a tree23"   and na :: "nat" assumes "\<And>n. \<lbrakk>compare ka p = LESS; full (Suc n) l\<rbrakk> \<Longrightarrow> dfull n (del ka l)"   and "\<And>n. \<lbrakk>compare ka p = EQUAL; full (Suc n) r\<rbrakk> \<Longrightarrow> dfull n (del (if_eq EQUAL None ka) r)"   and "\<And>n. \<lbrakk>compare ka p = GREATER; full (Suc n) r\<rbrakk> \<Longrightarrow> dfull n (del (if_eq GREATER None ka) r)"   and "full (Suc na) (Branch2 l p r)" shows "dfull na (del ka (Branch2 l p r))"
2: fixes a :: "'b set" assumes "path_components_of Y \<subseteq> {a}" shows "path_components_of X \<subseteq> {inv_into (path_components_of X) h a}"
distinct_15: fixes x31 :: "fm"   and x32 :: "fm"   and x4 :: "fm" shows "x31 OR x32 \<noteq> Neg x4"
children_eq2_h: fixes ptr' :: "(_) object_ptr" assumes "ptr' \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr" shows "|h \<turnstile> get_child_nodes ptr'|\<^sub>r = |h2 \<turnstile> get_child_nodes ptr'|\<^sub>r"
card_permutations_of_set: fixes A :: "'b set" assumes "finite A" shows "card (permutations_of_set A) = fact (card A)"
fds_deriv_times: fixes f :: "'a fds"   and g :: "'a fds" shows "fds_deriv (f * g) = fds_deriv f * g + f * fds_deriv g"
rcoset_replist_reorder: fixes H :: "'g set"   and gs :: "'g list"   and g :: "'g"   and gs' :: "'g list" assumes "is_rcoset_replist H (gs @ g # gs')" shows "is_rcoset_replist H (g # gs @ gs')"
pconj_def: fixes p :: "real"   and q :: "real" shows "p .& q \<equiv> p + q \<ominus> 1"
b_e_check_single_sound_binop_relop: fixes \<C> :: "t_context"   and e :: "b_e"   and tn' :: "checker_type"   and tm' :: "checker_type"   and t :: "t"   and iop :: "binop_i"   and fop :: "binop_f"   and irop :: "relop_i"   and frop :: "relop_f"   and tm :: "t list" assumes "check_single \<C> e tn' = tm'"   and "e = Binop_i t iop \<and> is_int_t t \<or> e = Binop_f t fop \<and> is_float_t t \<or> e = Relop_i t irop \<and> is_int_t t \<or> e = Relop_f t frop \<and> is_float_t t"   and "c_types_agree tm' tm"   and "tn' \<noteq> Bot"   and "tm' \<noteq> Bot" shows "\<exists>tn. c_types_agree tn' tn \<and> \<C> \<turnstile> [e] : tn _> tm"
add_vert_def: fixes v :: "'a" shows "add_vert v = \<lparr>verts = insert v (verts G), arcs = arcs G, tail = tail G, head = head G\<rparr>"
rt_strictly_fresher_singleI: fixes rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and i :: "nat"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "the (rt1 i) \<sqsubseteq> the (rt2 i)"   and "\<not> rt1 \<approx>\<^bsub>i\<^esub> rt2" shows "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"
Min_in: fixes A :: "'a DBMEntry set" assumes "finite A"   and "A \<noteq> {}" shows "linordered_monoid.Min A \<in> A"
length_fusion_def: fixes g :: "('d, 'e) generator"   and s :: "'e" shows "length_fusion g s = length (list.unfoldr g s)"
that: fixes x :: "'a" assumes "x extreme_point_of S" shows "thesis"
3: shows "exec_plan s as = exec_plan s (as1 @ as3)"
vertical_composite_axioms: shows "vertical_composite A C (K \<circ> F) (K \<circ> G) (K \<circ> H) (K \<circ> \<tau>) (K \<circ> \<tau>')"
proj_k_def: fixes x :: "nat" shows "proj_k x = outer_prod (ket_k x) (ket_k x)"
cases: fixes a1 :: "cld"   and a2 :: "fd list" assumes "class_fields a1 a2" obtains dcl :: "char list" and cl :: "cl" and fds :: "fd list" and meth_defs :: "meth_def list" where "a1 = cld_def dcl cl fds meth_defs" and "a2 = fds"
fst_apfst: fixes f :: "'e \<Rightarrow> 'c"   and x :: "'e \<times> 'd" shows "fst (apfst f x) = f (fst x)"
upper_closed: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and y :: "'e" assumes "connection G"   and "y \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "\<pi>\<^sub>*\<^bsub>G\<^esub> y \<in> carrier \<X>\<^bsub>G\<^esub>"
freeword_inject: fixes x :: "'a freeword"   and y :: "'a freeword" shows "(freeword x = freeword y) = (x = y)"
product_def: fixes A :: "('e, 'f, 'g) FSM"   and B :: "('e, 'f, 'h) FSM" shows "product A B \<equiv> \<lparr>succ = \<lambda>a (p\<^sub>1, p\<^sub>2). succ A a p\<^sub>1 \<times> succ B a p\<^sub>2, inputs = inputs A \<union> inputs B, outputs = outputs A \<union> outputs B, initial = (initial A, initial B)\<rparr>"
Iagree_Pred: fixes I :: "('d, 'e, 'f) interp"   and J :: "('d, 'e, 'f) interp"   and V :: "('d + 'e + 'f) set"   and p :: "'f" assumes "Iagree I J V"   and "Inr (Inr p) \<in> V" shows "Predicates I p = Predicates J p"
less_eq_bit1_def: fixes a :: "'a bit1"   and b :: "'a bit1" shows "(a \<le> b) = (Rep_bit1 a \<le> Rep_bit1 b)"
bij_to_def: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set" shows "bij_to f A B = (surj_to f A B \<and> inj_on f A)"
mult_1_left: fixes a :: "'a" shows "1\<^sub>\<pi> \<parallel> a = a"
prv_dsj_com_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (dsj \<phi> \<chi>) (dsj \<chi> \<phi>))"
Union_of_def: fixes f :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "Union_of f A \<equiv> \<Union> (f ` A)"