hcju/isabelleps huggingface dataset:
map_append_alt_def: fixes f :: "'b \<Rightarrow> 'a"   and b :: "'a list"   and xs :: "'b list" shows "map_append f b xs = map f xs @ b"
Map_comp': fixes t' :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and t :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "seq t' t" shows "Map (t' \<cdot> t) = vertical_composite.map (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Map t) (Map t')"
step: fixes x :: "name"   and xs :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "inputChain (x # xs) N P = \<nu>xinputChain xs N P"
simps: fixes a1 :: "('a, 'b) Term.term"   and a2 :: "('a, 'b) Term.term" shows "(a1 \<le>\<cdot> a2) = (\<exists>t s \<sigma>. a1 = s \<and> a2 = t \<and> t = subst_atm_abbrev s \<sigma>)"
approx_slp_appr_def: fixes fas :: "floatarith list"   and slp :: "floatarith list"   and X :: "real list set" shows "approx_slp_appr fas slp X = approx_slp_spec fas DIM('c) slp X \<bind> (\<lambda>cfp. case cfp of None \<Rightarrow> SUCCEED | Some cfp \<Rightarrow> RETURN (eucl_of_list ` cfp ::: appr_rel))"
assms_3: shows "j\<^sub>2 = encode f\<^sub>2"
prems_2: shows "unforwarded_non_volatile_reads (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (r # sb)) W \<subseteq> A"
proc_poss_def: fixes p :: "lambda" shows "proc_poss p = PC ` (lambdas p \<times> maps_over (labels p) UNIV) \<union> PP ` prims p \<union> {AStop}"
properL_update: fixes cl :: "('a, 'b, 'c) cmd list"   and c' :: "('a, 'b, 'c) cmd"   and n :: "nat" assumes "properL cl"   and "proper c'" shows "properL (cl[n := c'])"
refines: shows "(\<rightarrow>) \<le> ord\<inverse>\<inverse>"
fold_random_permutation_fold: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "finite A" shows "fold_random_permutation f x A = map_pmf (\<lambda>xs. fold f xs x) (pmf_of_set (permutations_of_set A))"
T_measure2_def: fixes Q :: "'v \<Rightarrow> 'a"   and \<pi> :: "'v \<Rightarrow> 'v option" shows "T_measure2 Q \<pi> \<equiv> if Q r = \<infinity> then T_measure1 (A Q \<pi>) else card (nodes rg)"
content_box_cbox: fixes a :: "'a"   and b :: "'a" shows "content (box a b) = content (cbox a b)"
transp_relcompp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "transp r = (r OO r \<le> r)"
simps_3: fixes ODE1 :: "('h + 'j, 'i) ODE"   and ODE2 :: "('h + 'j, 'i) ODE"   and \<sigma> :: "'j \<Rightarrow> ('h, 'i) trm" shows "OsubstFO (OProd ODE1 ODE2) \<sigma> = OProd (OsubstFO ODE1 \<sigma>) (OsubstFO ODE2 \<sigma>)"
gSubstGVar2_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSubstGVar2 MOD \<equiv> \<forall>ys y Y' Y. wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<longrightarrow> gSubst MOD ys Y' Y y (Var ys y) (gVar MOD ys y) = Y"
sbelow_halfspaces_def: fixes sctns :: "'c sctn set" shows "sbelow_halfspaces sctns = \<Inter> (sbelow_halfspace ` sctns)"
extendFalsum: fixes S :: "('a, 'b) sequent"   and \<Gamma> :: "('a, 'b) form multiset"   and \<Delta> :: "('a, 'b) form multiset" assumes "extend S ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "ff \<in># \<Gamma>"
even_odd_interpolate_filterlim: fixes u :: "nat \<Rightarrow> 'c"   and F :: "'c filter"   and v :: "nat \<Rightarrow> 'c" shows "(filterlim u F sequentially \<and> filterlim v F sequentially) = filterlim (even_odd_interpolate u v) F sequentially"
restrict0_restrict0: fixes X :: "'c set"   and Y :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "restrict0 X (restrict0 Y f) = restrict0 (X \<inter> Y) f"
bnd_2: shows "v x \<le>\<^sub>u\<^sub>b \<B>\<^sub>u s x"
assms_2: fixes i :: "nat" assumes "\<forall>i. f i \<in> A"   and "\<forall>i. g i \<in> A"   and "size (g i) < size (f i)"   and "\<forall>j<i. f j = g j" shows "Q"
post_rule': fixes P :: "assn"   and f :: "'b Heap"   and Q :: "'b \<Rightarrow> assn"   and R :: "'b \<Rightarrow> assn" assumes "<P> f <Q>"   and "\<not> <P> f <R>" shows "\<exists>x. \<not> (Q x \<Longrightarrow>\<^sub>A R x)"
b_e_type_drop: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Drop" shows "\<exists>t. ts = ts' @ [t]"
vec_cnj_mult_mv: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and x :: "complex \<times> complex" shows "vec_cnj (A *\<^sub>m\<^sub>v x) = mat_cnj A *\<^sub>m\<^sub>v vec_cnj x"
scc_root_def: fixes E :: "('a \<times> 'a) set"   and s :: "('a, 'b) state_scheme"   and v :: "'a"   and scc :: "'a set" shows "scc_root' E s v scc = (is_scc E scc \<and> v \<in> scc \<and> v \<in> dom (discovered s) \<and> scc \<inter> dom (discovered s) \<subseteq> (tree_edges s)\<^sup>* `` {v})"
d_delta_lnexp_cf6_nonpos: fixes x :: "real" shows "diff_delta_lnexp_cf6 x \<le> 0"
converse_rtranclp_into_equivclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d" assumes "r\<^sup>*\<^sup>* y x" shows "equivclp r x y"
u_simps_3: shows "trg (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = src ((f \<star> g) \<star> h)"
is_expansion_MSLCons: fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and C :: "'a"   and b :: "real \<Rightarrow> real"   and e :: "real"   and basis :: "(real \<Rightarrow> real) list" assumes "is_expansion_aux xs (\<lambda>x. f x - eval C x * b x powr e) (b # basis)"   and "is_expansion C basis"   and "\<And>e'. e < e' \<Longrightarrow> f \<in> o(\<lambda>x. b x powr e')"   and "ms_exp_gt e (ms_aux_hd_exp xs)" shows "is_expansion_aux (MSLCons (C, e) xs) f (b # basis)"
pathVertices_fwd: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "isPath u p v" shows "pathVertices u p = pathVertices_fwd u p"
Im_Ln_pos_le: fixes z :: "complex" assumes "z \<noteq> 0" shows "(0 \<le> Im (Ln z) \<and> Im (Ln z) \<le> pi) = (0 \<le> Im z)"
simps_2: fixes a :: "nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and y :: "nat" shows "PrimRecOp1 a h (Suc y) = h y (PrimRecOp1 a h y)"
lsumr_rsuml_right_gpv_left_gpv: fixes gpv :: "('h, 'j, 'm) gpv" shows "map_gpv' id lsumr rsuml (right_gpv (left_gpv gpv)) = left_gpv (right_gpv gpv)"
find_closest_pair_dist: fixes ps :: "(int \<times> int) list"   and C\<^sub>0 :: "int \<times> int"   and C\<^sub>1 :: "int \<times> int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "sorted_snd ps"   and "(C\<^sub>0, C\<^sub>1) = find_closest_pair (c\<^sub>0, c\<^sub>1) ps" shows "sparse (dist (case C\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (set ps)"
csol_2: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "{t0--t} \<subseteq> T"
intro: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool" assumes "rpo_basis ground_heads_var gt_sym arity_sym arity_var extf" shows "rpo ground_heads_var gt_sym extf arity_sym arity_var"
BAD_iff: fixes f :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "(f \<in> mbs.BAD A P) = ((\<forall>i. f i \<in> A) \<and> bad P f)"
axioms_1: fixes R :: "('c, 'd) ring_scheme" assumes "principal_domain R" shows "domain R"
scut_trancl: fixes p :: "'state"   and q :: "'state" assumes "scut\<^sup>+\<^sup>+ p q" obtains w :: "'action list" where "path w p" and "target w p = q" and "set w \<inter> cuts = {}" and "w \<noteq> []"
less_eq_pair_def: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" shows "(a \<le> b) = (a = b \<or> a < b)"
linear_order_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
count_list_eq_0_iff: fixes xs :: "'b list"   and x :: "'b" shows "(count_list xs x = 0) = (x \<notin> set xs)"
on_curvep_infinity: fixes a :: "int"   and b :: "int"   and x :: "int"   and y :: "int" shows "on_curvep a b (x, y, \<zero>) = (x \<in> carrier R \<and> y \<in> carrier R)"
subst_implies_prv_S: fixes \<phi> :: "'fmla" shows "let SS = \<lambda>t1 t2. psubst S [(t1, xx), (t2, yy)] in \<phi> \<in> fmla \<longrightarrow> Fvars \<phi> = {xx} \<longrightarrow> bprv (SS \<langle>\<phi>\<rangle> \<langle>subst \<phi> \<langle>\<phi>\<rangle> xx\<rangle>)"
index_smult_mat_3: fixes a :: "'c"   and A :: "'c mat" shows "dim_col (a \<cdot>\<^sub>m A) = dim_col A"
assms: shows "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"
assms_1: shows "\<forall>i. \<exists>j\<ge>i. d \<le> dur xs j - dur xs i"
id_eq_obj_1: fixes v :: "i"   and x :: "\<nu>" shows "[x\<^sup>P \<^bold>= x\<^sup>P in v]"
simps_5: fixes p :: "nat"   and f :: "float"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Num f) vs = Some (num_aform (real_of_float f), 0)"
ZassenhausTr2: fixes H :: "'a set"   and H1 :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "\<natural>H \<triangleright> H1" shows "H1 \<diamondop> H \<inter> K = H \<inter> K \<diamondop> H1"
nat_filtration_from_eq_sets: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "sets (infinite_coin_toss_space.nat_filtration N n) = sets (nat_filtration n)"
AbruptIs_4: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and x :: "stmt" shows "G\<turnstile>(Some xc, s) \<midarrow>x\<rightarrow> (Some xc, s)"
simps_2: fixes f :: "'a \<times> 'c \<Rightarrow> 'a \<times> 'b"   and kv :: "'a \<times> 'c"   and xs :: "('a \<times> 'c) list" shows "map_pair f (kv # xs) = (let (k, v) = f kv; aux = map_pair f xs in if v = (0::'b) then aux else (k, v) # aux)"
tiling_Diff1E: fixes t :: "'a set"   and a :: "'a set"   and A :: "'a set set" assumes "t - a \<in> tiling A"   and "a \<in> A"   and "a \<subseteq> t" shows "t \<in> tiling A"
r_result_def: shows "r_result \<equiv> Cn 3 r_ifz [Cn 3 r_pdec1 [r_leap], Cn 3 r_pdec2 [r_leap], r_constn 2 0]"
inj_on_iff_eq_card: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A" shows "inj_on f A = (card (f ` A) = card A)"
star_sim: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y \<le> z \<cdot> x" shows "x \<cdot> y\<^sup>\<star> \<le> z\<^sup>\<star> \<cdot> x"
smallest_dist_exists: fixes M :: "'b set"   and h :: "'b" assumes "convex M"   and "closed M"   and "M \<noteq> {}" shows "\<exists>k. is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) k"
length_dominant_term_ms_aux: fixes xs :: "('a \<times> real) msllist" shows "length (snd (dominant_term_ms_aux xs)) = Suc (expansion_level TYPE('a))"
h_def: shows "h \<equiv> \<lambda>x. - g x + f x + f x * ln (b * x) powr (- e / 2) + f x * g x"
list2FWpolicys_eq_el: fixes Filter :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and p :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "Filter \<noteq> []" shows "Cp (list2policyR Filter) p = Cp (list2FWpolicy (rev Filter)) p"
hpsafe_Loop_simps: fixes a :: "('a, 'b, 'c) hp" shows "hpsafe a** = hpsafe a"
simps_2: fixes ks :: "int list"   and l :: "int"   and xs :: "nat list" shows "eval_pf (Le ks l) xs = (eval_dioph ks xs \<le> l)"
assms_3: shows "l \<notin> LID\<^sub>G s\<^sub>1"
inducts: fixes x :: "'b formula"   and P :: "'b formula \<Rightarrow> bool" assumes "x \<in> AXH"   and "\<And>F G. P (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F)"   and "\<And>F G. P ((F \<^bold>\<rightarrow> F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> G)"   and "\<And>F G H. P ((F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H)"   and "\<And>G H F. P ((G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H)" shows "P x"
map2_val_compat'D3: fixes f :: "('a \<times> 'e) list \<times> 'o \<Rightarrow> ('a \<times> 'f) list \<times> 'o"   and zs :: "('a \<times> 'e) list \<times> 'o" assumes "map2_val_compat' f"   and "oalist_inv zs" shows "fst ` set (fst (f zs)) \<subseteq> fst ` set (fst zs)"
cdim_UNIV_ell2: shows "cdim UNIV = CARD('b)"
LT20: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and B :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and D :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (A \<leadsto> B \<or> D) \<longrightarrow> (B \<leadsto> D) \<longrightarrow> A \<leadsto> D"
integrable_const: fixes c :: "'b"   and a :: "'a"   and b :: "'a" shows "(\<lambda>x. c) integrable_on cbox a b"
IH2: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "Vagree \<nu> \<omega> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO ODE2}. FVT (\<sigma> y))"   and "osafe ODE2" shows "ODE_sem (adjointFO I \<sigma> \<nu>) ODE2 = ODE_sem (adjointFO I \<sigma> \<omega>) ODE2"
minus_in_Ints_iff: fixes x :: "'a" shows "(- x \<in> \<int>) = (x \<in> \<int>)"
Rep_euclidean_space_inverse: fixes x :: "'d euclidean_space" shows "Abs_euclidean_space (Rep_euclidean_space x) = x"
dist_rieman_sphere_zero: fixes M :: "riemann_sphere" shows "dist_riemann_sphere' M M = 0"
determ_additiveD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "determ t" shows "Transformers.additive t"
senior_states_finite: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "finite (semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n)"
in_nlists_UNIV: fixes xs :: "'c list"   and n :: "nat" shows "(xs \<in> nlists UNIV n) = (length xs = n)"
X_2: shows "(f \<circ> cv) ` [X]\<^bsup>n\<^esup> \<subseteq> {i}"
distinct_295: fixes x151 :: "floatarith"   and x152 :: "nat"   and x16 :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Floor x16"
Crypt: fixes X :: "msg"   and H :: "msg set"   and K :: "key" assumes "X \<in> synth H"   and "Key K \<in> H" shows "Crypt K X \<in> synth H"
sel_neg: shows "\<forall>x. sel (cl_ecl x) \<subseteq> cl_ecl x \<and> (\<forall>y\<in>sel (cl_ecl x). negative_literal y)"
ordertype_nat_\<omega>: fixes N :: "nat set" assumes "infinite N" shows "ordertype N less_than = \<omega>"
iso_in_homE: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<cong> b\<guillemotright>" obtains "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" and "C.iso f"
anr: shows "ANR S \<and> ANR T \<or> ANR U"
index_smult_mat_2: fixes a :: "'a"   and A :: "'a mat" shows "dim_row (a \<cdot>\<^sub>m A) = dim_row A"
remove: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "x \<in> A" shows "sum g A = g x + sum g (A - {x})"
inj_term_simps_3: fixes v :: "Term.var" shows "\<langle>v\<rangle> = In2 v"
power2_eq_1_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> '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.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(power.power one times a 2 = one) = (a = one \<or> a = uminus one)"
bij_betw_sequence_of: fixes enum :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw enum {0..<card A} A" shows "bij_betw (sequence_of A enum) (A \<rightarrow>\<^sub>E B) {xs. set xs \<subseteq> B \<and> length xs = card A}"
doc_tokens_length: fixes p :: "(symbol \<times> character list) list" assumes "doc_tokens p" shows "length (chars p) \<le> length Doc"
tsubst_preserves_free: fixes \<theta> :: "('a, 'b) trm"   and \<sigma> :: "('a, 'c, 'b) subst" assumes "dfree \<theta>"   and "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'" shows "dfree (Tsubst \<theta> \<sigma>)"
ideD_3: fixes a :: "'c" assumes "Cop.ide a" shows "Cop.cod a = a"
g_def: shows "g \<equiv> \<lambda>x. if x \<in> S then 0::'b else f x"
all_orientable_characterisations_2: shows "(\<forall>x. orientable_1 x) = (\<forall>x. orientable_3 x)"
len_ts: shows "length (pps_to_list S) = dim_col A"
ts_invD: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> bool"   and I :: "'h \<Rightarrow> 'g option"   and ts :: "'h \<Rightarrow> ('i \<times> 'k \<Rightarrow>f nat) option"   and m :: "'j"   and t :: "'h"   and x :: "'i"   and ln :: "'k \<Rightarrow>f nat" assumes "ts_inv P I ts m"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "\<exists>i. I t = \<lfloor>i\<rfloor> \<and> P i t x m"
add_mset_eq_single: fixes b :: "'b"   and M :: "'b multiset"   and a :: "'b" shows "(b, M = a, {#}) = (b = a \<and> M = {#})"
head_arev: fixes a :: "'b" assumes "a \<in> arcs G" shows "head G (edge_rev M a) = tail G a"
below_monicE: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "below_monic f"   and "f x \<sqsubseteq> f y" shows "x \<sqsubseteq> y"
Hpr_0: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and t :: "'e"   and R' :: "('e set, 'h) Ring_scheme"   and Y :: "'e set"   and f :: "'e"   and g :: "'e"   and h :: "'e" shows "Hpr\<^bsub> R S X t R' Y f g h\<^esub> 0 = (g, h)"
prj_yields_pullback: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "limit_cone cospan_shape.comp (\<cdot>) (cospan_diagram.map (\<cdot>) f g) (local.dom (prj1 f g)) (cospan_diagram.mkCone (\<cdot>) f g (prj1 f g) (prj0 f g))"
parCompose: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and R :: "pi"   and Rel' :: "(pi \<times> pi) set"   and S :: "pi"   and Rel'' :: "(pi \<times> pi) set" assumes "P \<leadsto><Rel> Q"   and "R \<leadsto><Rel'> S"   and "(P, Q) \<in> Rel"   and "(R, S) \<in> Rel'"   and "\<And>P' Q' R' S'. \<lbrakk>(P', Q') \<in> Rel; (R', S') \<in> Rel'\<rbrakk> \<Longrightarrow> (P' \<parallel> R', Q' \<parallel> S') \<in> Rel''"   and "\<And>T U x. (T, U) \<in> Rel'' \<Longrightarrow> (<\<nu>x>T, <\<nu>x>U) \<in> Rel''" shows "P \<parallel> R \<leadsto><Rel''> Q \<parallel> S"
os_list_none: shows "emp \<Longrightarrow>\<^sub>A os_list [] None"
strong_induct: fixes P :: "'v \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'v"   and ty :: "ty" assumes "\<And>list z. P z (PR list)"   and "\<And>ty z. (\<And>z. P z ty) \<Longrightarrow> P z (NOT ty)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 AND ty2)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 OR ty2)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 IMP ty2)" shows "P z ty"
Inf_from_def: fixes Inf :: "'a inference set"   and N :: "'a set" shows "inference_system.Inf_from Inf N = {\<iota> \<in> Inf. set (prems_of \<iota>) \<subseteq> N}"
g: fixes Z :: "'a set" assumes "openin (top_of_set (g ` T)) Z" shows "openin (top_of_set T) (T \<inter> g -` Z)"
sup_absorb2: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "sup x y = y"
is_processT: fixes P :: "'b process" shows "([], {}) \<in> F P \<and> (\<forall>s X. (s, X) \<in> F P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<in> F P \<longrightarrow> (s, {}) \<in> F P) \<and> (\<forall>s X Y. (s, Y) \<in> F P \<and> X \<subseteq> Y \<longrightarrow> (s, X) \<in> F P) \<and> (\<forall>s X Y. (s, X) \<in> F P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> F P) \<longrightarrow> (s, X \<union> Y) \<in> F P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> F P \<longrightarrow> (s, X - {tick}) \<in> F P) \<and> (\<forall>s t. s \<in> D P \<and> tickFree s \<and> front_tickFree t \<longrightarrow> s @ t \<in> D P) \<and> (\<forall>s X. s \<in> D P \<longrightarrow> (s, X) \<in> F P) \<and> (\<forall>s. s @ [tick] \<in> D P \<longrightarrow> s \<in> D P)"
assms: shows "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) p) (x # y # zs)"
\<omega>_simps_1: shows "arr \<r>\<^sup>-\<^sup>1[r]"
polyn_Hom_coeff_to_coeff: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y"   and "pol_coeff S c" shows "pol_coeff B (cf_h f c)"
conj_mono_right: fixes c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "(c\<^sub>0::'a::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'a::refinement_lattice)" shows "(d::'a::refinement_lattice) \<iinter> (c\<^sub>0::'a::refinement_lattice) \<sqsubseteq> d \<iinter> (c\<^sub>1::'a::refinement_lattice)"
dom_in_domains: fixes f :: "'a" assumes "B.domains f \<noteq> {}" shows "B.dom f \<in> B.domains f"
group_primfuns_def: shows "group_primfuns \<equiv> groupsort.group_sort fst merge_primfun"
int_simps_20: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (P \<and> P) = P"
exec1_drop_left: fixes P1 :: "instr list"   and P2 :: "instr list"   and i :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and n :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list" assumes "P1 @ P2 \<turnstile> (i, s, stk) \<rightarrow> (n, s', stk')"   and "size P1 \<le> i" shows "P2 \<turnstile> (i - size P1, s, stk) \<rightarrow> (n - size P1, s', stk')"
preserves_cod: fixes f :: "'a" assumes "arr f" shows "Span.cod (SPN f) = SPN (cod f)"
fresh_var_fresh: fixes e :: "exp" shows "atom (fresh_var e) \<sharp> e"
d_diff: fixes t :: "real" shows "delayed_safe_distance.d t = delayed_safe_distance.other.s t - ego2.s t"
simps_3: fixes uv :: "'a extT" shows "(uv \<le> extT.Top) = True"
Rep_\<I>_inverse: fixes x :: "('h, 'i) \<I>" shows "Abs_\<I> (Rep_\<I> x) = x"
IVT': fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and y :: "'c"   and b :: "'d" assumes "f a \<le> y"   and "y \<le> f b"   and "a \<le> b"   and "continuous_on {a..b} f" shows "\<exists>x\<ge>a. x \<le> b \<and> f x = y"
mask_Suc_exp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "mask (Suc n) = or (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) (mask n)"
square_free_m_smultD: fixes m :: "int"   and c :: "int"   and f :: "int poly" assumes "poly_mod.square_free_m m (smult c f)" shows "poly_mod.square_free_m m f"
singular_chain_def: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_chain p X c \<equiv> Poly_Mapping.keys c \<subseteq> singular_simplex_set p X"
simps_1: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uu :: "'c"   and B :: "'c"   and uv :: "'d" assumes "concrete_category Obj Hom Id Comp" shows "Cod (MkArr uu B uv) = B"