hcju/isabelleps huggingface dataset:
weak_control_dependence_def: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g"   and n :: "'g"   and n' :: "'g" assumes "StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit" shows "StrongPostdomination.weak_control_dependence sourcenode targetnode valid_edge Exit n n' = (\<exists>a a' as. n' \<notin> set (CFG.sourcenodes sourcenode (a # as)) \<and> CFG.path sourcenode targetnode valid_edge n (a # as) n' \<and> StrongPostdomination.strong_postdominate sourcenode targetnode valid_edge Exit n' (targetnode a) \<and> valid_edge a' \<and> sourcenode a' = n \<and> \<not> StrongPostdomination.strong_postdominate sourcenode targetnode valid_edge Exit n' (targetnode a'))"
card_UNIV_finite_5_def: shows "card_UNIV_class.card_UNIV = Phantom(Enum.finite_5) 5"
lt_lookup_vectorize: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "punit.lt (lookup (vectorize_poly p) (component_of_term (lt p))) = lp p"
alw: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F" shows "\<turnstile> \<box>F"
A3: fixes l' :: "real"   and r' :: "real"   and ll :: "real"   and r :: "real" assumes "1 \<le> l'"   and "1 \<le> r'"   and "1 \<le> ll"   and "1 \<le> r" shows "\<alpha> * (l' + r') * (l' + ll) powr \<beta> * (r' + r) powr \<beta> \<le> (l' + r') powr \<beta> * (l' + ll + r') powr \<beta> * (l' + ll + r' + r)"
forget_1: fixes x :: "name"   and M :: "trm"   and c :: "coname"   and P :: "trm" assumes "x \<sharp> M" shows "M{x:=<c>.P} = M"
map_graph_selectors_2: fixes f :: "('f \<times> 'e) set"   and G :: "('d, 'f) labeled_graph" shows "edges (map_graph f G) = on_triple f `` edges G"
lessThan_iff: fixes i :: "'a"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a" shows "(i \<in> ord.lessThan less k) = less i k"
open_not_negligible: fixes S :: "'c set" assumes "open S"   and "S \<noteq> {}" shows "\<not> negligible S"
igSubstAllIPresIGWlsAllSTR_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSubstAllIPresIGWlsAllSTR MOD \<equiv> igSubstIPresIGWlsSTR MOD \<and> igSubstAbsIPresIGWlsAbsSTR MOD"
Ext: fixes x :: "ZF"   and y :: "ZF" shows "(x = y) = (\<forall>z. Elem z x = Elem z y)"
edges_are_in_V_2: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^bsub>G'\<^esub> w" shows "w \<in> V\<^bsub>G'\<^esub>"
locate_pred_def: fixes f :: "'a \<Rightarrow> nat"   and al :: "'a list"   and i :: "nat"   and n_j :: "nat \<times> nat" shows "locate_pred f al i n_j \<equiv> fst n_j < length al \<and> snd n_j < f (al ! fst n_j) \<and> i = lsum f (take (fst n_j) al) + snd n_j"
indmhomTr1: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and m :: "'a" assumes "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M" shows "(f\<^sup>\<flat>\<^bsub>R M, N\<^esub>) (m \<uplus>\<^bsub>M\<^esub> ker\<^bsub>M,N\<^esub> f) = f m"
int_less_mono: fixes a :: "nat"   and b :: "nat" assumes "a < b" shows "int a < int b"
Im1_quat_of_real: fixes a :: "real" shows "Im1 (quat_of_real a) = 0"
map_simp: fixes f :: "'a" assumes "R.arr f" shows "R.map f = f"
of_nat_mod: 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 m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "semiring_1.of_nat one plus zero (m mod n) = modulo (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
merge_index: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and I :: "nat set"   and J :: "nat set"   and n :: "nat"   and tI :: "'b option list"   and tJ :: "'b option list"   and t :: "'b option list"   and i :: "nat" assumes "getIJ getIJ"   and "I \<inter> J = {}"   and "wf_tuple n I tI"   and "wf_tuple n J tJ"   and "t = New_max.merge tI tJ"   and "i < n" shows "i \<in> I \<and> t ! i = tI ! i \<or> i \<in> J \<and> t ! i = tJ ! i \<or> i \<notin> I \<and> i \<notin> J \<and> t ! i = None"
SREF: shows "(s, s') \<in> \<langle>Rk\<rangle>Rs"
action_obs_def: fixes E :: "('b \<times> ('a, 'b) obs_event action) llist"   and a :: "nat" shows "action_obs E a = snd (lnth E a)"
simps_10: fixes f1 :: "'d \<Rightarrow> 'b"   and f2 :: "'e \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'e" shows "map_sp\<^sub>\<mu> f1 f2 (Put x21 x22) = Put (f1 x21) (f2 x22)"
antilinearI: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>c x. f (c *\<^sub>C x) = cnj c *\<^sub>C f x" shows "antilinear f"
x_eq: shows "x = (\<lambda>i. integral s (f i))"
Recv: shows "ev = Recv i' (occurs_on ev) r u u' m"
null_setsD1: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> null_sets M" shows "emeasure M A = 0"
power2_eq_square: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "x [^]\<^bsub>R\<^esub> 2 = x \<otimes>\<^bsub>R\<^esub> x"
index_map_mat_3: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b mat" shows "dim_col (map_mat f A) = dim_col A"
Ps_def: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "Ps cf \<equiv> T.prob cf {bT \<in> space (T cf). S cf bT}"
game_B_bound: shows "spmf game_B True \<le> local.hash.\<epsilon>_uh * real q * real q"
I: fixes ys :: "letter list list"   and w :: "letter list"   and zs :: "letter list list" assumes "bar ys"   and "T a (w # xs) zs"   and "T b ys zs" shows "bar zs"
eventually_at_right_linorder: fixes b :: "'c"   and a :: "'c" assumes "b < a" shows "\<forall>\<^sub>F x in at_right b. x \<in> {b<..<a}"
loop_orientation_top_split: fixes y :: "'a" shows "(linear y \<and> antisymmetric y) = (y \<sqinter> y\<^sup>T = (1::'a) \<and> linear y)"
implies_not_e1: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>"
abs_prod_def: fixes x :: "'c \<times> 'd" shows "\<bar>x\<bar> = (\<bar>fst x\<bar>, \<bar>snd x\<bar>)"
lift_opt_simps_2: fixes v :: "'b"   and e :: "'a" shows "lift_opt (Some v) e = Inr v"
simps: fixes rep_key_order :: "'c \<Rightarrow> 'a key_order"   and ko :: "'c"   and xs :: "('a \<times> 'b) list"   and ox :: "'c" shows "oalist_raw.sort_oalist_aux rep_key_order ko (xs, ox) = (if ko = ox then xs else comparator.sort_oalist (key_compare (rep_key_order ko)) xs)"
strict_contour_is_diff: fixes B :: "'b set"   and y :: "'b"   and Pr :: "('b \<times> 'b) set" shows "{a \<in> B. a \<succeq>[Pr] y \<and> (y, a) \<notin> Pr} = at_least_as_good y B Pr - as_good_as y B Pr"
list_all2_takeI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and n :: "nat" assumes "list_all2 P xs ys" shows "list_all2 P (take n xs) (take n ys)"
assms_2: shows "P \<Longrightarrow>\<^sub>A F * X"
of_int_0: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 0 = zero"
periodic_arg_plus_distrib: fixes x :: "complex"   and m :: "int"   and n :: "int" shows "x + of_int (m + n) = x + of_int n + of_int m"
dom_simp: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_BxA.arr f" shows "A_BxA.dom f = (A_B.dom (fst f), A.dom (snd f))"
Rep_G: fixes x :: "'b G" shows "Rep_G x \<in> G"
inner_minus_right: fixes x :: "'a"   and y :: "'a" shows "inner x (- y) = - inner x y"
paths_both: fixes xs :: "'b list"   and t :: "'b ttree"   and t' :: "'b ttree" shows "(xs \<in> paths (t \<otimes>\<otimes> t')) = (\<exists>ys\<in>paths t. \<exists>zs\<in>paths t'. xs \<in> ys \<otimes> zs)"
simps_7: fixes b :: "expr"   and e1 :: "expr"   and e2 :: "expr" shows "free_vars (IF b THEN e1 ELSE e2) = free_vars b \<union> free_vars e1 \<union> free_vars e2"
Collect_imp_eq: fixes P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" shows "{x. P x \<longrightarrow> Q x} = - {x. P x} \<union> {x. Q x}"
orderI: fixes a :: "'b"   and b :: "'b" assumes "a = max a b" shows "b \<le> a"
spmf_of_set: fixes A :: "'a set"   and x :: "'a" shows "spmf (spmf_of_set A) x = indicat_real A x / real (card A)"
pickInp_alphaInp_iff_equal: fixes inp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option"   and inp' :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" assumes "goodInp inp"   and "goodInp inp'" shows "((pickInp inp, pickInp inp') \<in> alphaInp) = (inp = inp')"
max_option_def: fixes pos :: "bool"   and o1 :: "int option"   and o2 :: "int option" shows "max_option pos o1 o2 = (if le_option pos o1 o2 then o2 else o1)"
cobounded2: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c" shows "less_eq_prod b (sup_prod a b)"
that: assumes "a \<^bold>\<in> A"   and "b \<^bold>\<in> B" shows "thesis"
subst_prm: fixes \<pi> :: "'a prm"   and M :: "'a trm"   and z :: "'a"   and N :: "'a trm" shows "\<pi> \<cdot> M[z ::= N] = \<pi> \<cdot> M[\<pi> $ z ::= \<pi> \<cdot> N]"
fdia_bbox_galois: fixes R :: "('d \<times> 'f) set"   and f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "(\<forall>x.  |R\<rangle> f x \<le> g x) = (\<forall>x. f x \<le> [R| g x)"
Extend_maximal: fixes f :: "nat \<Rightarrow> 'b fm"   and C :: "'b fm set set"   and S :: "'b fm set" assumes "\<forall>y. \<exists>n. y = f n"   and "finite_char C" shows "maximal (Extend S C f) C"
l1_obs_iagreement_Init: shows "oreach l1 \<subseteq> l1_iagreement_Init"
filter_sup_right_upper_bound: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b set"   and G :: "'b set" assumes "class.semilattice_inf inf less_eq less"   and "ord.filter less_eq F" shows "G \<subseteq> semilattice_inf.filter_sup inf less_eq F G"
Cons_eq_append_conv: fixes x :: "'g"   and xs :: "'g list"   and ys :: "'g list"   and zs :: "'g list" shows "(x # xs = ys @ zs) = (ys = [] \<and> x # xs = zs \<or> (\<exists>ys'. x # ys' = ys \<and> xs = ys' @ zs))"
simps_3: fixes v :: "'a"   and uv :: "('form, 'var) out_port" shows "hyps (Conclusion v) uv = None"
Cond_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and e' :: "'addr expr1"   and e'' :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (if (e) e' else e'') h s s'"
A: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "prob_space_with_indicators M I A" shows "A ` I \<subseteq> sets M"
bij_vimage_eq_inv_image: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set" assumes "bij f" shows "f -` A = inv f ` A"
lookup_Abs_poly_mapping: fixes f :: "'b \<Rightarrow> 'c" assumes "finite {x. f x \<noteq> (0::'c)}" shows "poly_mapping.lookup (Abs_poly_mapping f) = f"
subset_insert_iff: fixes A :: "'e set"   and x :: "'e"   and B :: "'e set" shows "(A \<subseteq> insert x B) = (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)"
\<pi>1: shows "product_cone J1 (\<cdot>) D1 a1 \<pi>1"
hyps: fixes l2 :: "'a list"   and w1 :: "'a list"   and w2 :: "'a list" assumes "l1' @ l2 \<in> w1 \<otimes> w2" shows "\<exists>w11 w12 w21 w22. w1 = w11 @ w12 \<and> w2 = w21 @ w22 \<and> l1' \<in> w11 \<otimes> w21 \<and> l2 \<in> w12 \<otimes> w22"
assms_1: shows "Simulation_Invariants A B sim PA QA PB QB"
close_nonmember_eq: fixes A :: "'a list"   and a :: "'a" assumes "\<not> List.member A a" shows "close A a = A"
agree_iff_subset: fixes G :: "('e, 'f) labeled_graph"   and X :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set"   and g :: "('f \<times> 'g) set" assumes "graph_homomorphism G X f"   and "univalent g" shows "agree_on G f g = (f \<subseteq> g)"
sq_norm_pos: fixes n :: "nat"   and fs :: "'a vec list"   and j :: "nat" assumes "gram_schmidt_fs_lin_indpt n fs"   and "j < length fs" shows "(0::'a) < \<parallel>gram_schmidt_fs.gso n fs j\<parallel>\<^sup>2"
select_convs_7: fixes types_t :: "tf list"   and func_t :: "tf list"   and global :: "tg list"   and table :: "nat option"   and memory :: "nat option"   and local :: "t list"   and label :: "t list list"   and return :: "t list option"   and more :: "'a" shows "label \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr> = label"
max_def: fixes a :: "'c"   and b :: "'c" shows "ord_class.max a b = (if a \<le> b then b else a)"
map\<^sub>0_simps_3: fixes a :: "('a E.term, 'a) arr" assumes "obj a" shows "trg\<^sub>B (DN.map\<^sub>0 a) = DN.map\<^sub>0 a"
permute_u_def: fixes p :: "perm"   and x :: "'c\<^sub>\<bottom>" shows "p \<bullet> x = (case x of up\<cdot>x \<Rightarrow> up\<cdot>(p \<bullet> x))"
preserves_obj: fixes a :: "'c" assumes "C.obj a" shows "obj (E a)"
cod_simps_2: shows "\<mu>\<nu>\<pi>.cod.leg0 = \<mu>\<nu>\<pi>.leg0"
simps_2: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and a :: "axiom"   and \<sigma> :: "('sf, 'sc, 'sz) subst" shows "step_result (SG, C) (i, AxSubst a \<sigma>) = (closeI SG i, C)"
Int_subset_iff: fixes C :: "'b set"   and A :: "'b set"   and B :: "'b set" shows "(C \<subseteq> A \<inter> B) = (C \<subseteq> A \<and> C \<subseteq> B)"
smap_un2: fixes X :: "'a set"   and y :: "'a" shows "smap (insert (smap X) (\<eta> y)) = smap (X \<union> \<eta> y)"
u_2: shows "s \<in> set (u \<leadsto> t)"
rangeE: fixes b :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "b \<in> range f" obtains x :: "'c" where "b = f x"
is_iso_class_def: fixes F :: "('a, 'a set) Maps.arr set" shows "Maps.is_iso_class F \<equiv> \<exists>f. f \<in> F \<and> F = Maps.iso_class f"
\<Z>_sgrpI: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_sgrp_basis \<alpha> \<SS>"   and "\<And>a b c. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c)" shows "\<Z>_sgrp \<alpha> \<SS>"
mset_map_id: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b"   and X :: "'a multiset" assumes "\<And>x. f (g x) = x" shows "f `# g `# X = X"
admissibleD: fixes lub :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and A :: "'b set" assumes "ccpo.admissible lub ord P"   and "Complete_Partial_Order.chain ord A"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> P x" shows "P (lub A)"
valid_def: fixes P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> ('e \<times> 'd) \<times> bool"   and Q :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace> \<equiv> \<forall>s. P s \<longrightarrow> (\<forall>r s'. (r, s') = fst (f s) \<longrightarrow> Q r s')"
trivial_D2: fixes \<sigma> :: "'a \<times> ('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and r :: "('b \<times> 'b) set" assumes "\<sigma> \<in> seq ars" shows "D2 r (\<sigma>, (fst \<sigma>, []), (lst \<sigma>, []), \<sigma>)"
conc_star_comm: fixes A :: "'b list set" shows "A @@ star A = star A @@ A"
card_insert_le: fixes A :: "'d set"   and x :: "'d" assumes "finite A" shows "card A \<le> card (insert x A)"
simps_1: fixes p :: "real" shows "lprob_comp p [] = 1"
nth_mem: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "xs ! n \<in> set xs"
less_two_power: fixes n :: "nat" shows "n < 2 ^ n"
prod_ord_rawI: fixes xs :: "('a \<times> 'd) list \<times> 'o"   and ys :: "('a \<times> 'e) list \<times> 'o"   and P :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool" assumes "oalist_inv xs"   and "oalist_inv ys"   and "\<And>k. k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys) \<Longrightarrow> P k (lookup_raw xs k) (lookup_raw ys k)" shows "prod_ord_raw P xs ys"
test_not: fixes p :: "'a" shows "\<iota> (- p) = ! (\<iota> p)"
plus_float_def: fixes a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "a + b = fadd To_nearest a b"
card_image_le: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i" assumes "finite A" shows "card (f ` A) \<le> card A"
IH_5: assumes "delete\<langle>t,s\<rangle> \<in> set (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<sigma>)" shows "\<exists>t' s'. t = t' \<cdot> \<sigma> \<and> s = s' \<cdot> \<sigma> \<and> delete\<langle>t',s'\<rangle> \<in> set S"
extf_ext_trans: fixes f :: "'s" shows "ext_trans (extf f)"
map_2: fixes f :: "'h \<Rightarrow> 'g"   and x21 :: "'h"   and x22 :: "'h list" shows "map f (x21 # x22) = f x21 # map f x22"
inject_2: fixes x21 :: "nat"   and x22 :: "nat"   and y21 :: "nat"   and y22 :: "nat" shows "(Operator x21 x22 = Operator y21 y22) = (x21 = y21 \<and> x22 = y22)"
fst_initmissing_netgmap_pair_fst: fixes s :: "((state \<times> 'a) \<times> 'b) net_state" shows "fst (initmissing (netgmap (\<lambda>(p, q). (fst (Fun.id p), snd (Fun.id p), q)) s)) = fst (initmissing (netgmap fst s))"
trg_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "span_vertical_category.trg C \<mu> \<equiv> if partial_magma.arr (span_vertical_category.vcomp C) \<mu> then \<lparr>Chn = partial_magma.dom C (partial_magma.cod C (Leg1 (Dom \<mu>))), Dom = \<lparr>Leg0 = partial_magma.cod C (Leg1 (Dom \<mu>)), Leg1 = partial_magma.cod C (Leg1 (Dom \<mu>))\<rparr>, Cod = \<lparr>Leg0 = partial_magma.cod C (Leg1 (Dom \<mu>)), Leg1 = partial_magma.cod C (Leg1 (Dom \<mu>))\<rparr>\<rparr> else partial_magma.null (span_vertical_category.vcomp C)"
monom_one: fixes R :: "('c, 'd) ring_scheme" assumes "UP_ring R" shows "monom (UP R) \<one>\<^bsub>R\<^esub> 0 = \<one>\<^bsub>UP R\<^esub>"
wit: shows "0 \<notin> set ks \<and> charpoly A = prod_root_unity ks * f \<and> charpoly (HMA_Connect.map_matrix complex_of_real A) = prod_root_unity ks * map_poly complex_of_real f \<and> (\<forall>x. poly (charpoly (HMA_Connect.map_matrix complex_of_real A)) x = 0 \<longrightarrow> cmod x \<le> 1) \<and> (\<forall>x. poly (map_poly complex_of_real f) x = 0 \<longrightarrow> cmod x < 1)"
mset_insert: fixes a :: "'b"   and h :: "'b heap" shows "mset_heap (insert a h) = {#a#} + mset_heap h"
elims: fixes xb :: "'o"   and x :: "('a \<times> 'c) list"   and xa :: "'a"   and y :: "'c" assumes "lookup_pair xb x xa = y" obtains xb :: "'a" where "x = []" and "xa = xb" and "y = (0::'c)"   | k :: "'a" and v :: "'c" and xs :: "('a \<times> 'c) list" and xc :: "'a" where "x = (k, v) # xs" and "xa = xc" and "y = (case key_compare (rep_key_order xb__) xc k of Eq \<Rightarrow> v | Lt \<Rightarrow> 0::'c | Gt \<Rightarrow> lookup_pair xb__ xs xc)"
subst_fm_id: fixes A :: "fm"   and a :: "name" shows "A(a::=Var a) = A"
f_newind: fixes anything :: "nat list" shows "{\<lambda>nl. nl = xs @ r # 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ r # 0 # 0 \<up> (fft - Suc far) @ anything}"
not_Some_eq: fixes x :: "'a option" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = None)"
gamm_cont: shows "continuous_on {0..1} (\<lambda>a. \<gamma> a \<bullet> i)"
PO_a0i_refines_init_a0n: shows "init a0i \<subseteq> R0n0i `` init a0n"
foo3D: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and ps :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and r :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "wellformed_policy1 p"   and "p = DenyAll # ps"   and "applied_rule_rev Cp x p = \<lfloor>DenyAll\<rfloor>"   and "r \<in> set ps" shows "x \<notin> dom (Cp r)"
asInp_equal_iff_alphaInp: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option"   and qinp' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" assumes "qGoodInp qinp" shows "(asInp qinp = asInp qinp') = ((qinp, qinp') \<in> alphaInp)"
hypnat_less_one: fixes n :: "nat star" shows "(n < 1) = (n = 0)"
le_imp_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.semi_canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" shows "\<exists>c. b = plus a c"
wf_dual_lang_dual_wf_word: fixes n :: "nat"   and r :: "'b rexp_dual" assumes "wf_dual n r" shows "\<forall>w\<in>lang_dual n r. wf_word n w"
list_rel_coeffs_minus_i: fixes x1 :: "'i list"   and x2 :: "'a list"   and y1 :: "'i list"   and y2 :: "'a list" assumes "list_all2 R x1 x2"   and "list_all2 R y1 y2" shows "list_all2 R (coeffs_minus_i ops x1 y1) (coeffs_minus x2 y2)"
assms: shows "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>r new_element_ptr"
group_hom_axioms_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" shows "group_hom_axioms G H h \<equiv> h \<in> hom G H"
almost_full_on_subset: fixes A :: "'c set"   and B :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "A \<subseteq> B"   and "almost_full_on P B" shows "almost_full_on P A"
pmdl_Macaulay_list: fixes ps :: "('t \<Rightarrow>\<^sub>0 'd) list" shows "pmdl (set (Macaulay_list ps)) = pmdl (set ps)"
wf_box_dmd_decrease: fixes r :: "('a \<times> 'a) set"   and v :: "state \<Rightarrow> 'a" assumes "wf r" shows "\<turnstile> \<box>\<diamond>(v$, $v) \<in> #r \<longrightarrow> \<box>\<diamond><(v$, $v) \<notin> #r>_v"
path_image_const: fixes a :: "'a" shows "path_image (\<lambda>t. a) = {a}"
simps_11: fixes n :: "nat"   and I :: "\<I>"   and \<phi> :: "Formula.formula" shows "minit0 n (formula.Next I \<phi>) = MNext I (minit0 n \<phi>) True []"
compact_space: fixes X :: "'b topology" shows "compact_space X = (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> \<Union> \<U> = topspace X \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> \<Union> \<F> = topspace X))"
type_copy_map_cong0: fixes M :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d"   and N :: "'e \<Rightarrow> 'b"   and h :: "'d \<Rightarrow> 'e"   and f :: "'b \<Rightarrow> 'f" assumes "M (g x) = N (h x)" shows "(f \<circ> M \<circ> g) x = (f \<circ> N \<circ> h) x"
key_gen_def: shows "key_gen = sample_uniform (order \<G>) \<bind> (\<lambda>x. return_spmf (\<^bold>g [^] x, x))"
SubobjsR_nonempty: fixes 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"   and C :: "char list"   and Cs :: "char list list" assumes "Subobjs\<^sub>R P C Cs" shows "Cs \<noteq> []"