hcju/isabelleps huggingface dataset:
vector_space_pair_ow_def: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f"   and scale\<^sub>1 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h"   and scale\<^sub>2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'h" shows "vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 \<and> vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
f: shows "\<forall>x\<in>s. (g has_vector_derivative D x) (at x)"
surj_to_def: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" shows "surj_to f A B = (f ` A = B)"
NFAND_conjuncts: fixes \<psi> :: "('a, 'b) aformula" assumes "nf_ACI \<psi>" shows "NFAND (conjuncts \<psi>) = \<psi>"
match_4: fixes g :: "'a2"   and f :: "'a2"   and h :: "'a2" assumes "A2.seq g f"   and "A2.seq h g" shows "A2.seq h (g \<cdot>\<^sub>A\<^sub>2 f)"
delta_remove: fixes S :: "'c set"   and a :: "'c"   and b :: "'c \<Rightarrow> 'a"   and c :: "'c \<Rightarrow> 'a" assumes "finite S" shows "F (\<lambda>k. if k = a then b k else c k) S = (if a \<in> S then b a \<^bold>* F c (S - {a}) else F c (S - {a}))"
min_int_poly_represents: fixes x :: "'a" assumes "algebraic x" shows "min_int_poly x represents x"
True: shows "p dvd f \<and> f \<noteq> 0"
sigma_algebra_sigma_sets: fixes a :: "'d set set"   and \<Omega> :: "'d set" assumes "a \<subseteq> Pow \<Omega>" shows "sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
cons3: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and es :: "exp0 list"   and s3 :: "'b state"   and err :: "v error_result" assumes "evaluate ck env s1 e (s2, Rval v1) \<and> evaluate_list ck env s2 es (s3, Rerr err)" shows "evaluate_list ck env s1 (e # es) (s3, Rerr err)"
not_infTp_I_intT: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "\<not> infTp \<sigma>" shows "I_intT \<sigma> a"
reflects_ide: fixes \<mu> :: "'a" assumes "C.endo \<mu>"   and "D.ide (F \<mu>)" shows "C.ide \<mu>"
Sup2_E: fixes A :: "('c \<Rightarrow> 'd \<Rightarrow> bool) set"   and a :: "'c"   and b :: "'d" assumes "Sup A a b" obtains r :: "'c \<Rightarrow> 'd \<Rightarrow> bool" where "r \<in> A" and "r a b"
is_faithful_functor: shows "faithful_functor (\<cdot>) C S.map"
bind_lzero: fixes f :: "'c \<Rightarrow> 'b option" shows "Option.bind None f = None"
minus_1_div_2_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.unique_euclidean_ring_with_nat divide plus minus zero times one modulo euclidean_size division_segment uminus" shows "divide (uminus one) (numeral.numeral one plus (num.Bit0 num.One)) = uminus one"
subst_all_mono: fixes M :: "('c, 'd) Term.term set"   and M' :: "('c, 'd) Term.term set"   and s :: "'d \<Rightarrow> ('c, 'e) Term.term" assumes "M \<subseteq> M'" shows "M \<cdot>\<^sub>s\<^sub>e\<^sub>t s \<subseteq> M' \<cdot>\<^sub>s\<^sub>e\<^sub>t s"
gb_sig_def: fixes rword_strict :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and fs0 :: "('a \<Rightarrow>\<^sub>0 'c) list" shows "gb_sig rword_strict fs0 = map snd (fst (gb_sig_z rword_strict fs0))"
distinct_11: fixes x3 :: "'a formula" shows "\<bottom> \<noteq> \<^bold>\<not> x3"
in_fset_eqvt: fixes p :: "perm"   and x :: "'b"   and S :: "'b fset" shows "p \<bullet> (x |\<in>| S) = (p \<bullet> x |\<in>| p \<bullet> S)"
flow_residual_add: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow_attainability \<Delta>"   and "countable_network.wf_residual_network \<Delta>"   and "flow \<Delta> f"   and "flow (countable_network.residual_network \<Delta> f) g" shows "flow \<Delta> (f \<oplus>\<^bsub>\<Delta>\<^esub> g)"
dbl_def: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus x = plus x x"
dim_kronecker_1: fixes A :: "'b mat"   and B :: "'b mat" shows "dim_row (kronecker_product A B) = dim_row A * dim_row B"
to_total_def: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and v :: "'Var" shows "to_total \<Gamma> v \<equiv> if v \<in> dom \<Gamma> then the (\<Gamma> v) else dma v"
zero_less_arctan_iff: fixes x :: "real" shows "(0 < arctan x) = (0 < x)"
max_y_def: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and j :: "nat" shows "hlde_ops.max_y a b x j = (if j < length b \<and> hlde_ops.Ej a b j x \<noteq> {} then Min (hlde_ops.Ej a b j x) else Max (set a))"
gen_step_eq_assert: fixes gds :: "('a, 'b) gen_dfs_struct"   and V0 :: "'a set"   and s :: "'b" assumes "gen_dfs_defs.gen_cond gds V0 s"   and "rwof (gds_init gds) (gen_dfs_defs.gen_cond gds V0) (gen_dfs_defs.gen_step gds V0) s" shows "gen_dfs_defs.gen_step gds V0 s = gen_dfs_defs.gen_step_assert gds V0 s"
challenge_space_def: shows "challenge_space = {..<order \<G>}"
basis_expansion: fixes x :: "('a, 'b) vec" shows "(\<Sum>i\<in>UNIV. x $ i *s axis i (1::'a)) = x"
assms_3: shows "x \<in> local.set (local.prod (local.dom f) (local.dom g))"
relabel_pres_Labeling: fixes u :: "nat" assumes "relabel_precond f l u" shows "Labeling c s t f (relabel_effect f l u)"
eq2: shows "inner_prod (beta_l l \<cdot>\<^sub>v \<beta>) (psi_l l) = beta_l l * beta_l l"
aNDDA: fixes p :: "('b set set, 'c) Combinators list" assumes "allNetsDistinct p" shows "allNetsDistinct (DenyAll # p)"
assms_2: shows "is_factorization_of (a, fctrs) p"
ord_le_eq_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a \<le>s b"   and "b = c" shows "a \<le>s c"
emptyrem2taut: fixes Cn :: "'b set \<Rightarrow> 'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "Compact_logic Cn"   and "Tarskian_logic.remainder Cn A \<phi> = {}" shows "\<phi> \<in> Cn {}"
statImpEnt: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Phi> :: "'c" assumes "\<Psi> \<hookrightarrow> \<Psi>'"   and "\<Psi> \<turnstile> \<Phi>" shows "\<Psi>' \<turnstile> \<Phi>"
card_Int_if_either_in: fixes S :: "'a set"   and T :: "'a set" assumes "\<forall>x\<in>S. \<forall>y\<in>S. x = y \<or> x \<notin> T \<or> y \<notin> T" shows "card (S \<inter> T) \<le> 1"
mat_of_col_dim_2: fixes v :: "'b vec" shows "dim_col (mat_of_col v) = 1"
sep_conj_empty': fixes P :: "'a \<Rightarrow> bool" shows "(\<box> \<and>* P) = P"
assms_2: shows "B \<Delta> C \<in> null_sets M"
simps: fixes x :: "'b ext"   and y :: "'b ext" shows "x * y = x \<sqinter> y"
terminal_path_is_path: fixes v :: "'vertex"   and v' :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "terminal_path v v' pth" shows "path v v' pth"
AAss_\<tau>red1t_xt3: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and a :: "'addr val"   and i :: "'addr val" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (Val a\<lfloor>Val i\<rceil> := e, xs) (Val a\<lfloor>Val i\<rceil> := e', xs')"
IMP_LIST_def: fixes A :: "bool list"   and B :: "bool" shows "IMP_LIST A B \<equiv> ALL_LIST A \<longrightarrow> B"
idiv_by_0: fixes a :: "enat" shows "a div 0 = 0"
multeqp_def: fixes P :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and N :: "'k multiset"   and M :: "'k multiset" shows "multeqp P N M = (let Z = M \<inter># N; X = M - Z; Y = N - Z in \<forall>y\<in>#Y. \<exists>x\<in>#X. P y x)"
le_mod_greater_imp_div_less: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a \<le> b"   and "b mod m < a mod m" shows "a div m < b div m"
Arcsin_Arccos_csqrt_pos: fixes z :: "complex" assumes "0 < Re z \<or> Re z = 0 \<and> 0 \<le> Im z" shows "Arcsin z = Arccos (csqrt (1 - z\<^sup>2))"
less_acom_def: fixes x :: "'a acom"   and y :: "'a acom" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
distinct_28: fixes x41 :: "trm"   and x42 :: "trm"   and x31 :: "POPLmarkRecord.type"   and x32 :: "trm" shows "x41 \<bullet> x42 \<noteq> (\<lambda><:x31. x32)"
valid_proj_neq_succ_restricted_neq_succ: fixes x' :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and vs :: "'d set"   and s :: "('d, 'e) fmap" assumes "x' \<in> prob_proj PROB vs"   and "state_succ s x' \<noteq> s" shows "fmrestrict_set vs (state_succ s x') \<noteq> fmrestrict_set vs s"
ASucFbd_Cinfinite: fixes p1 :: "('f \<times> 'f) set"   and r2a :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "Cinfinite ((p1 +c ctwo) ^c cardSuc ((bd_F1 +c r2a) +c r2))"
small_big_antisym: fixes h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "h1 \<in> G"   and "h2 \<in> G"   and "h1 \<in> o[F](h2)"   and "h2 \<in> O[F](h1)" shows "False"
balance_left_in_tree: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and a :: "'a"   and b :: "'b" assumes "inv1l l"   and "inv1 r"   and "bheight l + 1 = bheight r" shows "((k, v) \<in> set (entries (balance_left l a b r))) = ((k, v) \<in> set (entries l) \<or> k = a \<and> v = b \<or> (k, v) \<in> set (entries r))"
product_target: fixes w :: "('d \<times> 'e) list"   and r\<^sub>1 :: "'f list"   and r\<^sub>2 :: "'g list"   and p\<^sub>1 :: "'f"   and p\<^sub>2 :: "'g" assumes "length w = length r\<^sub>1"   and "length r\<^sub>1 = length r\<^sub>2" shows "target (w || r\<^sub>1 || r\<^sub>2) (p\<^sub>1, p\<^sub>2) = (target (w || r\<^sub>1) p\<^sub>1, target (w || r\<^sub>2) p\<^sub>2)"
l_inv_ex: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "\<exists>y\<in>carrier (K [X]). y \<oplus>\<^bsub>K [X]\<^esub> x = \<zero>\<^bsub>K [X]\<^esub>"
retraction_is_epi: fixes g :: "'a" assumes "C.retraction g" shows "C.epi g"
and_num_8388608_128: shows "8388608 AND 128 = 0"
comparable_cases: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "connex A less_eq"   and "x \<in> A"   and "y \<in> A" obtains "less_eq x y"   | "less_eq y x"
pdevs_val_of_list_of_pdevs: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and X :: "'a pdevs" assumes "e \<in> funcset UNIV I"   and "0 \<in> I" obtains e' :: "nat \<Rightarrow> real" where "pdevs_val e (pdevs_of_list (map snd (list_of_pdevs X))) = pdevs_val e' X" and "e' \<in> funcset UNIV I"
axiom5_F: fixes f1 :: "'o \<Rightarrow> 'r"   and f2 :: "'p \<Rightarrow> 'q" shows "F'set4 \<circ> Fmap f1 f2 = (`) f2 \<circ> F'set4"
sumSym: fixes P :: "pi"   and Q :: "pi" shows "P \<oplus> Q \<sim> Q \<oplus> P"
intY1_elem: fixes c :: "'a" assumes "c \<in> intY1" shows "c \<in> A"
IH: fixes cs :: "'edge list" assumes "upd_cs cs as' = c' # cs'" shows "upd_cs cs (as' @ [a]) = cs'"
ineqs_vars_not_bound: shows "\<forall>X F x. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<longrightarrow> x \<in> {x \<in> vars\<^sub>s\<^sub>t S. \<exists>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<and> x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<and> x \<notin> set X} \<longrightarrow> x \<notin> set X"
de_Morgan: fixes P :: "'b \<Rightarrow> bool" assumes "\<not> (\<forall>x. P x)" shows "\<exists>x. \<not> P x"
a: shows "snd (usubstappp \<sigma> U \<alpha>**) \<noteq> undefg"
Cons_1: fixes e :: "exp" assumes "set T \<subseteq> range Arg" shows "(\<Gamma>, eta_expand (length T) e, T @ S) \<Rightarrow>\<^sup>* (\<Gamma>, e, T @ S)"
sgn_le_0_iff: fixes x :: "real" shows "(sgn x \<le> 0) = (x \<le> 0)"
1: shows "\<guillemotleft>lF f : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright> \<and> F (trg\<^sub>C f) \<star>\<^sub>D lF f = (F \<i>\<^sub>C[trg\<^sub>C f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
t\<^sub>p_sumofALGreal: fixes s :: "'c list \<times> 'd"   and a :: "nat \<times> nat list"   and qs :: "'c list"   and i :: "nat" assumes "distinct (fst s)"   and "snd a = []"   and "qs ! i \<in> set (fst s)" shows "real (t\<^sub>p (fst s) (qs ! i) a) = (\<Sum>e\<in>set (fst s). real (ALG e qs i s))"
simps: fixes a1 :: "'a"   and a2 :: "'b"   and r :: "('a \<times> 'c) set"   and s :: "('c \<times> 'b) set" shows "((a1, a2) \<in> r O s) = (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r \<and> (b, c) \<in> s)"
inject_2: fixes x21 :: "'d \<times> 'e"   and x22 :: "('d, 'e, 'f) ctree fset"   and y21 :: "'d \<times> 'e"   and y22 :: "('d, 'e, 'f) ctree fset" shows "(cNode x21 x22 = cNode y21 y22) = (x21 = y21 \<and> x22 = y22)"
af_prefix_false: fixes i :: "nat"   and j :: "nat"   and \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "i \<le> j"   and "af \<phi> (prefix i w) \<sim>\<^sub>P false\<^sub>n" shows "af \<phi> (prefix j w) \<sim>\<^sub>P false\<^sub>n"
1: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and oldDefs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and oldUses :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and phis :: "'a \<Rightarrow> 'b \<times> 'e \<Rightarrow> 'e list option"   and var :: "'a \<Rightarrow> 'e \<Rightarrow> 'd"   and g :: "'a"   and P :: "'e set" assumes "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var"   and "CFG_SSA_Transformed.redundant_set \<alpha>n inEdges' defs uses phis g P" shows "\<exists>scc\<subseteq>P. CFG_SSA_Transformed.redundant_scc \<alpha>n inEdges' defs uses phis g P scc"
isFaultE: fixes s :: "('d, 'e) xstate" assumes "isFault s" obtains f :: "'e" where "s = Fault f"
hta_ensure_idx_f_correct_\<alpha>: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_\<alpha> (hta_ensure_idx_f H) = hta_\<alpha> H"
cases: fixes a1 :: "(char list \<times> pat) list"   and a2 :: "(char list \<times> POPLmarkRecord.type) list"   and a3 :: "binding list" assumes "\<turnstile> a1 [:] a2 \<Rightarrow> a3" obtains "a1 = []" and "a2 = []" and "a3 = []"   | p :: "pat" and T :: "POPLmarkRecord.type" and \<Delta>\<^sub>1 :: "binding list" and fps :: "(char list \<times> pat) list" and fTs :: "(char list \<times> POPLmarkRecord.type) list" and \<Delta>\<^sub>2 :: "binding list" and l :: "char list" where "a1 = (l, p) \<Colon> fps" and "a2 = (l, T) \<Colon> fTs" and "a3 = \<up>\<^sub>e \<parallel>\<Delta>\<^sub>1\<parallel> 0 \<Delta>\<^sub>2 @ \<Delta>\<^sub>1" and "\<turnstile> p : T \<Rightarrow> \<Delta>\<^sub>1" and "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>\<^sub>2" and "fps\<langle>l\<rangle>\<^sub>? = \<bottom>"
linear_cf_axioms: fixes d :: "'d" shows "linear_cf (Pd d) (Cd d)"
map_minsky_def: fixes f :: "'i \<Rightarrow> 'g"   and g :: "'j \<Rightarrow> 'h"   and M :: "('i \<times> ('i, 'j) Op) set" shows "map_minsky f g M = map_prod f (map_Op f g) ` M"
finite_reach: fixes \<phi> :: "'a ltl" shows "finite (reach \<Sigma> (delta \<Sigma>) (initial \<phi>))"
homomorphism_chain_boundary: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "chain_boundary p \<in> hom (relcycle_group p X S) (relcycle_group (p - Suc 0) (subtopology X S) {})"
966: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - - (x \<sqinter> y)) = top"
conj_xor_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<^bold>\<sqinter> (y \<oplus> z) = x \<^bold>\<sqinter> y \<oplus> x \<^bold>\<sqinter> z"
maddux_3_13: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y) \<sqinter> - x = y \<sqinter> - x"
lin_indpt_list_def: fixes fs :: "'a vec list" shows "lin_indpt_list fs = (set fs \<subseteq> carrier_vec n \<and> distinct fs \<and> lin_indpt (set fs))"
feasible_Cons: fixes c :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and ls :: "('c, 'd) label list" shows "feasible c (l # ls) = (\<exists>c'. se c l c' \<and> Conf.sat c' \<and> feasible c' ls)"
token_run_step: fixes x :: "nat"   and n :: "nat"   and q' :: "'a ltl_prop_equiv_quotient"   and q :: "'a ltl_prop_equiv_quotient" assumes "x \<le> n"   and "\<UU>.token_run x n = q'"   and "q = \<up>af\<^sub>G\<^sub>\<UU> q' (w n)" shows "\<UU>.token_run x (Suc n) = q"
freshTransition_1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P'"   and "c \<sharp> P"   and "c \<noteq> x" shows "c \<sharp> P'"
3: shows "times_top_class.total (top * (v \<sqinter> - F * e * top \<sqinter> top * e\<^sup>T * F))"
pos_mult_l_gt: fixes w :: "int"   and i :: "int"   and j :: "int" assumes "0 < w"   and "i \<le> j"   and "0 \<le> i" shows "i \<le> w * j"
atLeastatMost_empty_iff2: fixes a :: "'k multiset"   and b :: "'k multiset" shows "({} = subset_mset.atLeastAtMost a b) = (\<not> a \<subseteq># b)"
simps_2: fixes v :: "'c" shows "Num_arc_delta v + MinInfty_delta = Num_arc_delta v"
max_in_subcomplex: fixes Y :: "'b set set"   and y :: "'b set" assumes "order.greater_eq Y Y \<and> SimplicialComplex Y"   and "y \<in> Y"   and "codomain.chamber y" shows "SimplicialComplex.maxsimp Y y"
new_msgs: shows "msgs (S t i) cid = Marker # msgs (S t (i + 1)) cid"
iMOD_inext_nth: fixes r :: "nat"   and m :: "nat"   and a :: "nat" shows "[ r, mod m ] \<rightarrow> a = r + m * a"
le_Err: fixes e :: "'b err"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "e \<sqsubseteq>\<^bsub>le r\<^esub> Err"
E_Proj: fixes t :: "trm"   and t' :: "trm"   and l :: "char list" assumes "t \<longmapsto> t'" shows "t..l \<longmapsto> t'..l"
trivial_relative_homology_group_gen: fixes X :: "'a topology"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and p :: "int" assumes "continuous_map X (subtopology X S) f"   and "homotopic_with (\<lambda>h. True) (subtopology X S) (subtopology X S) f id"   and "homotopic_with (\<lambda>k. True) X X f id" shows "trivial_group (relative_homology_group p X S)"
legacy_Complex_simps_4: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" shows "Complex a b - Complex c d = Complex (a - c) (b - d)"
join_return: fixes xs :: "'a\<cdot>'b" shows "join\<cdot>(return\<cdot>xs) = xs"
discr_Sbis: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and d :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "proper c"   and "proper d"   and "PL_Indis.discr aval tval cval indis c"   and "PL_Indis.discr aval tval cval indis d" shows "(c, d) \<in> PL_Indis.Sbis aval tval cval indis"
oalist_inv_raw_singleton: fixes k :: "'a"   and v :: "'c" shows "oalist_inv_raw [(k, v)] = (v \<noteq> (0::'c))"
list_def: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "list s n = map s [0..<n]"
is_valid_problem_sas_plus_then_strips_transformation_too_v: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>x. ((\<phi> \<Psi> )\<^sub>G) x \<noteq> None \<longrightarrow> ListMem x ((\<phi> \<Psi> )\<^sub>\<V>)"
RPred1_altdef: shows "\<R>\<^sub>0\<^sub>1 = {f \<in> \<R>. \<forall>x. the (f x) \<le> 1}"
assms_4: shows "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"
carrier_matI: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "dim_row A = nr"   and "dim_col A = nc" shows "A \<in> carrier_mat nr nc"
Dflow_def: fixes x0 :: "'a" shows "Y.Dflow x0 = Y.mvar.flow x0 0 1\<^sub>L"
lemParallelReflexive: fixes lineA :: "'a Line" shows "lineA \<parallel>  lineA"
emb_step_at_right_context: fixes p :: "dir list"   and dir :: "dir"   and t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "emb_step_at (dir.Right # p) dir (App t s) = App t (emb_step_at p dir s)"
add_mset: fixes x :: "'a"   and N :: "'a multiset" shows "\<Sum>\<^sub># (add_mset x N) = x + \<Sum>\<^sub># N"
SN_subset: fixes R :: "('b \<times> 'b) set"   and R' :: "('b \<times> 'b) set" assumes "SN R"   and "R' \<subseteq> R" shows "SN R'"
cop__one_or_two_sides: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D"   and "\<not> Col C A B"   and "\<not> Col D A B" shows "A B TS C D \<or> A B OS C D"
unit_disc_fix_preserve_poincare_line: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v" shows "poincare_line (moebius_pt M u) (moebius_pt M v) = moebius_circline M (poincare_line u v)"
3: fixes b :: "ereal"   and a :: "ereal" assumes "b \<le> a" shows "interval_lebesgue_integral lborel a b f = - interval_lebesgue_integral lborel b a f"
lookup_empty: fixes k :: "'e" shows "Mapping.lookup Mapping.empty k = None"
chamber_shared_facet: fixes C :: "'a set"   and z :: "'a set"   and D :: "'a set" assumes "chamber C"   and "z \<lhd> C"   and "D \<in> X"   and "z \<lhd> D" shows "chamber D"
simps_25: fixes vc :: "'a"   and vd :: "bexp"   and ve :: "'a acom"   and vf :: "'a"   and v :: "char list"   and va :: "aexp"   and vb :: "'a" shows "({vc} WHILE vd DO ve {vf} \<le> v ::= va {vb}) = False"
Red_preserves_iconf: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"   and "iconf (shp s) e" shows "iconf (shp s') e'"
rep_eq: fixes p :: "'b pmf"   and s :: "'b set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "measure_pmf (cond_pmf p s) = uniform_measure (measure_pmf p) s"
disjointI: fixes A :: "'b set set" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; a \<noteq> b\<rbrakk> \<Longrightarrow> a \<inter> b = {}" shows "disjoint A"
unit_in_vhom: shows "\<guillemotleft>D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) : src\<^sub>D (F f) \<Rightarrow>\<^sub>D F g \<star>\<^sub>D F f\<guillemotright>"