hcju/isabelleps huggingface dataset:
fvt: fixes t :: "'trm" assumes "set (x # map snd txs) \<subseteq> var"   and "set (Var y # t # map fst txs) \<subseteq> trm"   and "set [] \<subseteq> fmla"   and "t \<in> set (Var y # t # map fst txs)" shows "set (getFrN (x # map snd txs) (Var y # t # map fst txs) [] (Suc (length txs))) \<inter> FvarsT t = {}"
ImageI: fixes a :: "'h"   and b :: "'i"   and r :: "('h \<times> 'i) set"   and A :: "'h set" assumes "(a, b) \<in> r"   and "a \<in> A" shows "b \<in> r `` A"
lessThan_Int_lessThan: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.linorder less_eq less" shows "ord.greaterThan less a \<inter> ord.greaterThan less b = ord.greaterThan less (ord.max less_eq a b)"
image_comp: fixes f :: "'b::type \<Rightarrow> 'a::type"   and g :: "'c::type \<Rightarrow> 'b::type"   and r :: "'c::type set" shows "(f::'b::type \<Rightarrow> 'a::type) ` (g::'c::type \<Rightarrow> 'b::type) ` (r::'c::type set) = (f \<circ> g) ` r"
yoneda_functor_fixed_object_axioms_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" shows "yoneda_functor_fixed_object_axioms C a \<equiv> partial_magma.ide C a"
Cignorabimus_union: fixes M :: "variable set"   and \<alpha> :: "game"   and V :: "variable set"   and N :: "variable set" assumes "M \<in> Cignorabimus \<alpha> V"   and "N \<in> Cignorabimus \<alpha> V" shows "M \<union> N \<in> Cignorabimus \<alpha> V"
derivative_quotient_bound_left: fixes g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and K :: "real"   and t :: "real" assumes "(g has_vderiv_on g') {a..b}"   and "\<And>t. t \<in> {a..b} \<Longrightarrow> K \<le> g' t / g t"   and "continuous_on {a..b} g'"   and "\<And>t. t \<in> {a..b} \<Longrightarrow> 0 < g t"   and "t \<in> {a..b}" shows "g t \<le> g b * exp (K * (t - b))"
similarity_def: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" shows "similarity A M = mat_inv A *\<^sub>m\<^sub>m M *\<^sub>m\<^sub>m A"
tendsto_mult_ereal_PInf: fixes f :: "'b \<Rightarrow> ereal"   and l :: "ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal" assumes "(f \<longlongrightarrow> l) F"   and "0 < l"   and "(g \<longlongrightarrow> \<infinity>) F" shows "((\<lambda>x. f x * g x) \<longlongrightarrow> \<infinity>) F"
distinct_129: fixes x10 :: "nat" shows "DBff \<noteq> DBNum x10"
T_quasi_preserves_null2_1: fixes A :: "'a set" assumes "A \<in> null_sets M" shows "T --` A \<in> null_sets M"
span_mem: fixes E :: "'c set"   and u :: "'c" assumes "E \<subseteq> carrier V"   and "u \<in> E" shows "u \<in> local.module.span E"
exec_step_ind_InitThrow: fixes C :: "char list"   and Cs :: "char list list"   and a :: "nat"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepT (C # Cs) a) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Throwing Cs a) # frs, sh(C \<mapsto> (fst (the (sh C)), Error)))"
type_less_x_Bag: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> < Bag \<sigma>" obtains \<rho> :: "'a OCL_Types.type" where "\<tau> = Bag \<rho>" and "\<rho> < \<sigma>"
cfg_on_def: fixes s :: "'s" shows "cfg_on s = {cfg. cfg_onp s cfg}"
measurable_embed_measure2: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "inj f" shows "f \<in> M \<rightarrow>\<^sub>M embed_measure M f"
measurable_on_compose_continuous_box: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and g :: "'b \<Rightarrow> 'c" assumes "f measurable_on UNIV"   and "\<And>x. f x \<in> box a b"   and "continuous_on (box a b) g" shows "g \<circ> f measurable_on UNIV"
rel_map_1: fixes Sb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'c \<Rightarrow> 'a"   and x :: "'c option"   and y :: "'b option" shows "rel_option Sb (map_option i x) y = rel_option (\<lambda>x. Sb (i x)) x y"
kerQmp: fixes H :: "'a set"   and N :: "'a set" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N" shows "ker\<^bsub>M /\<^sub>m H,M /\<^sub>m N\<^esub> (Mp\<^bsub>M  H,N\<^esub>) = carrier (mdl M N /\<^sub>m H)"
UN_upper: fixes a :: "'b"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "a \<in> A" shows "order.greater_eq (\<Union> (B ` A)) (B a)"
pair_graph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs G"   and "w \<notin> pverts G" shows "pair_graph (subdivide G e w)"
Pxz: shows "distributed M (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) (\<lambda>x. ennreal (Pxz x))"
sec_list_def: fixes x :: "'a list" shows "sec x = length x"
negligible_convex_frontier: fixes S :: "'a set" assumes "convex S" shows "negligible (frontier S)"
claim: fixes A :: "bool"   and B :: "bool"   and C :: "bool" shows "(A \<or> B \<or> C) = (\<not> C \<longrightarrow> \<not> B \<longrightarrow> A)"
i_less_length: shows "i < length [0..<nrows_iarray (matrix_to_iarray A)]"
cInf_greatest: fixes X :: "'c multiset set"   and z :: "'c multiset" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "z \<subseteq># Inf X"
Class_widen_Class: fixes P :: "'b prog"   and C :: "char list"   and D :: "char list" shows "P \<turnstile> Class C \<le> Class D = P \<turnstile> C \<preceq>\<^sup>* D"
not_None_eq: fixes x :: "'b option" shows "(x \<noteq> \<bottom>) = (\<exists>y. x = \<lfloor>y\<rfloor>)"
sconcat_sfoldl: shows "sconcat = sfoldl\<cdot>sappend\<cdot>[::]"
prognosis_Var_thunk: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and u :: "Arity"   and as :: "Arity list"   and S :: "stack_elem list"   and a :: "Arity" assumes "map_of \<Gamma> x = Some e"   and "ae x = up\<cdot>u"   and "\<not> isVal e" shows "prognosis ae as u (delete x \<Gamma>, e, Upd x # S) \<sqsubseteq> record_call x\<cdot>(prognosis ae as a (\<Gamma>, Var x, S))"
generalized_fw_joinD: fixes f1 :: "('a simple_match \<times> 'b) list"   and f2 :: "('a simple_match \<times> 'c) list"   and p :: "('a, 'd) simple_packet_scheme"   and u :: "'a simple_match"   and d1 :: "'b"   and d2 :: "'c" assumes "generalized_sfw (generalized_fw_join f1 f2) p = Some (u, d1, d2)" shows "\<exists>r1 r2. generalized_sfw f1 p = Some (r1, d1) \<and> generalized_sfw f2 p = Some (r2, d2) \<and> Some u = simple_match_and r1 r2"
simps: fixes a1 :: "'b set list"   and a2 :: "'b"   and a3 :: "'b" shows "of_weak_ranking a1 a2 a3 = (\<exists>i j xs x y. a1 = xs \<and> a2 = y \<and> a3 = x \<and> i \<le> j \<and> i < length xs \<and> j < length xs \<and> x \<in> xs ! i \<and> y \<in> xs ! j)"
range_disjointed_sets: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and A :: "nat \<Rightarrow> 'c set" assumes "ring_of_sets \<Omega> M"   and "range A \<subseteq> M" shows "range (disjointed A) \<subseteq> M"
mult_def: fixes r :: "('c \<times> 'c) set" shows "mult r = (mult1 r)\<^sup>+"
stable_on_allocation: fixes ds :: "'b set"   and X :: "'a set" assumes "PdXXX.stable_on ds X" shows "inj_on Xd X"
keys_replace: fixes k :: "'a"   and v :: "'b"   and m :: "('a, 'b) mapping" shows "Mapping.keys (Mapping.replace k v m) = Mapping.keys m"
tl_append2: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs \<noteq> []" shows "tl (xs @ ys) = tl xs @ ys"
pac_ideal_def: fixes A :: "int mpoly set" shows "pac_ideal A \<equiv> More_Modules.ideal (A \<union> polynomial_bool)"
extensionalD_insert: fixes a :: "'a"   and A :: "'a set"   and d :: "'b"   and B :: "'b set" assumes "a \<notin> A" shows "extensionalD d (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f\<in>extensionalD d A \<inter> (A \<rightarrow> B). fun_upd f a ` B)"
pos_cost: fixes e :: "'b" assumes "e \<in> arcs G" shows "0 \<le> c e"
llexord_trans: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" assumes "llexord r xs ys"   and "llexord r ys zs"   and "\<And>a b c. \<lbrakk>r a b; r b c\<rbrakk> \<Longrightarrow> r a c" shows "llexord r xs zs"
reflexivity: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" assumes "preference carrier relation" shows "refl_on carrier relation"
skl1_inv1_trans: shows "{skl1_inv1} TS.trans skl1 {> skl1_inv1}"
assms_4: fixes B :: "'a set" assumes "B \<in># \<B>" shows "K_GDD (B \<times> I) (f B) \<K>' {{x} \<times> I |x. x \<in> B}"
assms_2: shows "seq \<p>\<^sub>1[f, g] h"
Collect_disj_eq: fixes P :: "'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool" shows "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}"
Nullstellensatz: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "finite X"   and "F \<subseteq> P[X]"   and "f \<in> \<I> (\<V> F)" shows "f \<in> \<surd>ideal F"
name_per_rev: fixes pi :: "(name \<times> name) list"   and x :: "'a" shows "pi \<bullet> rev pi \<bullet> x = x"
Q_implies_path: fixes I :: "'a set"   and A :: "'a set"   and a :: "'a"   and b :: "'a" assumes "I \<subseteq> A"   and "A \<in> \<P>"   and "Q I a b" shows "b \<in> A \<and> a \<in> A"
uadmit_mkv_ntadjoint: fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and ODE :: "('sf + 'b, 'sz) ODE" assumes "\<And>i. dsafe (\<sigma> i)"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO ODE}. FVT (\<sigma> y))"   and "osafe ODE" shows "mk_v (adjointFO I \<sigma> \<nu>) ODE = mk_v (adjointFO I \<sigma> \<omega>) ODE"
mlex_less: fixes f :: "'a \<Rightarrow> nat"   and x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set" assumes "f x < f y" shows "(x, y) \<in> f <*mlex*> R"
is_ground_subst_atm_list: fixes As :: "'a list"   and \<sigma> :: "'s" assumes "is_ground_atm_list As" shows "As \<cdot>al \<sigma> = As"
reflexion_preserves_finiteness: fixes P1 :: "'a eclause"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "finite (cl_ecl P1)"   and "reflexion P1 C \<sigma> k C'" shows "finite (cl_ecl C) \<and> finite C'"
pgallery_def: fixes xs :: "'a set list" shows "pgallery xs \<equiv> Ball (set xs) chamber \<and> padjacentchain xs"
lossless_iff_pmf_None: fixes p :: "'a spmf" shows "lossless_spmf p = (pmf p None = 0)"
card_equiv_rel_eq_card_partitions: fixes A :: "'b set" shows "card {R. equiv A R} = card {P. partition_on A P}"
assms_7: shows "moebius_pt M2 z1 = w1"
trancl_mono_mp: fixes U :: "('c \<times> 'c) set"   and V :: "('c \<times> 'c) set"   and x :: "'c \<times> 'c" assumes "U \<subseteq> V"   and "x \<in> U\<^sup>+" shows "x \<in> V\<^sup>+"
ranI: fixes m :: "'f \<Rightarrow> 'e option"   and a :: "'f"   and b :: "'e" assumes "m a = Some b" shows "b \<in> ran m"
mono_generate_ring: fixes I :: "'a set"   and J :: "'a set" assumes "I \<subseteq> J"   and "J \<subseteq> carrier R" shows "generate_ring R I \<subseteq> generate_ring R J"
trnl\<^sub>\<epsilon>_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b"   and u :: "'b"   and \<nu> :: "'b" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "adjunction_in_bicategory.trnl\<^sub>\<epsilon> V H \<a> \<i> src trg f g \<epsilon> u \<nu> \<equiv> V (bicategory.lunit V H \<a> \<i> src trg u) (V (H \<epsilon> u) (V (bicategory.\<a>' V H \<a> src trg f g u) (H f \<nu>)))"
closure_same_affine_hull: fixes S :: "'b set" shows "affine hull closure S = affine hull S"
weakly_bisimilar_eqvt: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and Q :: "'a"   and p :: "perm" assumes "weak_nominal_ts satisfies transition \<tau>"   and "weak_nominal_ts.weakly_bisimilar satisfies transition \<tau> P Q" shows "weak_nominal_ts.weakly_bisimilar satisfies transition \<tau> (p \<bullet> P) (p \<bullet> Q)"
f: shows "finite {a \<in> A. f a \<noteq> (0::'b)}"
regular_one_closed: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b" assumes "class.stone_relation_algebra inf less_eq less sup bot top uminus one times conv" shows "one = uminus (uminus one)"
isRefV_simps_2: fixes i :: "int" shows "isRefV (intgV i) = False"
cupcake_list_complete: fixes env :: "v sem_env"   and s1 :: "'b state"   and es :: "exp0 list"   and res :: "'b state \<times> (v list, v) result" assumes "Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s1 es res"   and "is_cupcake_all_env env"   and "list_all is_cupcake_exp es" shows "\<exists>rs. list_all2_shortcircuit (cupcake_evaluate_single env) es rs \<and> sequence_result rs = snd res"
simps_1: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'c" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
i_def: shows "i = Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))"
simps_4: fixes x :: "'a word"   and y :: "'a word" shows "proper_interval (Some x) (Some y) = (x < y \<and> x \<noteq> y - 1)"
eval_sees_field_i_i_i_o_o_conv: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and F :: "char list" shows "pred.eval (sees_field_i_i_i_o_o P C F) = (\<lambda>(T, D). P \<turnstile> C sees F:T in D)"
finite_simplex: fixes x :: "'a set" assumes "x \<in> X" shows "finite x"
zero_le_divide_1_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide one a) = less_eq zero a"
nfa_reach_subset_memb: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and q0 :: "bool list"   and bss :: "bool list list"   and q :: "bool list" assumes "subset_DFS A n"   and "nfa_reach A q0 bss q"   and "nfa_is_node A q0"   and "list_all (is_alph n) bss" shows "subset_memb q (subset_dfs A q0)"
G: shows "xs @ zs = xs' @ drop (length xs') (xs @ zs)"
zero_closed: shows "\<zero>\<^bsub>K [X]\<^esub> \<in> carrier (K [X])"
P_conforms: shows "path_conforms_with_strategy p P' \<sigma>'"
assms_2: shows "\<rho> f|` S \<sqsubseteq> \<rho>' f|` S"
zmod_zminus2_eq_if: fixes a :: "int"   and b :: "int" shows "a mod - b = (if a mod b = 0 then 0 else a mod b - b)"
valid_node_def: fixes n :: "'b LDCFG_node" shows "pd'.valid_node n \<equiv> \<exists>a. lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a \<and> (n = src a \<or> n = trg a)"
chamber_preimage: fixes C :: "'a set" assumes "C \<in> X"   and "folding.chamber (f ` C)" shows "folding.chamber C"
cts_def_2: shows "ct_list_eq cts (to_ct_list bs)"
SN_ars_def: fixes S :: "('a \<times> 'a) set" shows "SN_ars S \<equiv> SN S"
if_split_asm: fixes P :: "'a \<Rightarrow> bool"   and Q :: "bool"   and x :: "'a"   and y :: "'a" shows "P (if Q then x else y) = (\<not> (Q \<and> \<not> P x \<or> \<not> Q \<and> \<not> P y))"
trancl_rtrancl_trancl: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set"   and z :: "'c" assumes "(x, y) \<in> r\<^sup>+"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>+"
sgn2_pos_eq_one: fixes x :: "nat" assumes "0 < sgn2 x" shows "sgn2 x = 1"
simps: fixes I :: "'a set"   and v :: "nat \<Rightarrow> 'b"   and as :: "('a \<times> 'b atom) set" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as = v \<Turnstile>\<^sub>a\<^sub>s (snd ` (as \<inter> I \<times> UNIV))"
assms: shows "run_one_step d i (s, vs, ves, $Grow_memory) = (s', vs', res)"
line_integral_rec_join_cons: fixes F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set"   and \<gamma> :: "real \<Rightarrow> real \<times> real"   and l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" assumes "line_integral_exists F basis \<gamma>"   and "line_integral_exists F basis (rec_join l)"   and "\<And>k' \<gamma>'. (k', \<gamma>') \<in> set ((1, \<gamma>) # l) \<Longrightarrow> valid_path \<gamma>'"   and "finite basis" shows "line_integral F basis (rec_join ((1, \<gamma>) # l)) = line_integral F basis (\<gamma> +++ rec_join l)"
aux0_1: fixes xs :: "('b, 'c) Combinators list"   and x :: "('b, 'c) Combinators" assumes "singleCombinators xs"   and "singleCombinators [x]" shows "singleCombinators (x # xs)"
is_equivalence_functor: shows "equivalence_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F"
subringE_2: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "subring H R" shows "\<zero>\<^bsub>R\<^esub> \<in> H"
negligible_Un: fixes S :: "'a set"   and T :: "'a set" assumes "negligible S"   and "negligible T" shows "negligible (S \<union> T)"
converse_trancl_induct: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (y, b) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(y, z) \<in> r; (z, b) \<in> r\<^sup>+; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
strict_trans: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a dvd b \<and> a \<noteq> b"   and "b dvd c \<and> b \<noteq> c" shows "a dvd c \<and> a \<noteq> c"
int_ops_9: fixes a :: "nat"   and b :: "nat" shows "int (a mod b) = int a mod int b"
K_def: shows "K = (\<lambda>n. {x \<in> space M. \<bar>f x\<bar> \<in> {a * L * m n<..<a * L * m (Suc n)}})"
has_field_mono: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list"   and C' :: "char list" assumes "P \<turnstile> C has F,b:T in D"   and "P \<turnstile> C' \<preceq>\<^sup>* C" shows "P \<turnstile> C' has F,b:T in D"
assms_6: shows "f ` (sphere (0::'a) 1 \<inter> S) \<subseteq> sphere (0::'a) 1 \<inter> T"
strong_induct: fixes P :: "'a \<Rightarrow> dbtm \<Rightarrow> bool"   and c :: "'a"   and dbtm :: "dbtm" assumes "\<And>c. P c DBZero"   and "\<And>x c. P c (DBVar x)"   and "\<And>x c. P c (DBInd x)"   and "\<And>x1 x2a c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2a\<rbrakk> \<Longrightarrow> P c (DBEats x1 x2a)" shows "P c dbtm"
Basic: fixes a :: "'c set"   and A :: "'c set set"   and sp :: "'c set" assumes "a \<in> A" shows "a \<in> sigma_sets sp A"
succ_neq_zero: fixes x :: "V" shows "ZFC_in_HOL.succ x \<noteq> 0"
mono: fixes x :: "nat"   and y :: "nat" assumes "P x y" shows "x \<le> y"
scc_root_is_discovered: fixes v :: "'v"   and scc :: "'v set" assumes "scc_root s' v scc" shows "v \<in> dom (discovered s')"
nf_inres_RES: fixes X :: "'a set"   and x :: "'a" shows "nf_inres (RES X) x = (x \<in> X)"
intro: fixes V :: "'a set"   and smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<exists>ms. set ms \<subseteq> V \<and> R_scalar_mult.RSpan UNIV smult ms = V" shows "FinDimVectorSpace_axioms smult V"
insert: fixes x :: "'a"   and A :: "'a set" shows "MaxR_opt (insert x A) = (if x \<in> Field r then Some (case MaxR_opt A of None \<Rightarrow> x | Some y \<Rightarrow> maxR x y) else MaxR_opt A)"
coeff_smult: fixes a :: "'a"   and p :: "'a poly"   and n :: "nat" shows "coeff (Polynomial.smult a p) n = a * coeff p n"
isomorphicI': fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<cong> b\<guillemotright>" shows "isomorphic a b"
cr_gen_cfg_wf_def: shows "cr_gen_cfg_wf \<equiv> \<lambda>x y. x = Rep_gen_cfg_wf y"
realpow_pos_nth_unique: fixes n :: "nat"   and a :: "real" assumes "0 < n"   and "0 < a" shows "\<exists>!r. 0 < r \<and> r ^ n = a"
intros_2: fixes biG :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)"   and biI :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)"   and biH :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)"   and u :: "'a"   and v :: "'a"   and w :: "'a"   and uv :: "'b"   and uw :: "'b"   and vw :: "'b" assumes "subdivision biG biI"   and "subdivision_step (fst biI) (snd biI) (fst biH) (snd biH) (u, v, w) (uv, uw, vw)" shows "subdivision biG biH"
simps_9: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'c \<Rightarrow> 'b" shows "(case dFAILi of dSUCCEEDi \<Rightarrow> f1 | dFAILi \<Rightarrow> f2 | dRETURN x \<Rightarrow> f3 x) = f2"
norm_diff_triangle_ineq: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "norm (a + b - (c + d)) \<le> norm (a - c) + norm (b - d)"
msub_def: fixes m :: "int"   and x :: "int"   and y :: "int" shows "x --\<^bsub>m\<^esub> y = (x - y) mod m"
delta': fixes a :: "'b"   and g :: "'b \<Rightarrow> 'a" shows "(\<Prod>b. if a = b then g b else (1::'a)) = g a"
fmrel_fmdom'_eq: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'g \<rightharpoonup>\<^sub>f 'e"   and y :: "'g \<rightharpoonup>\<^sub>f 'f" assumes "fmrel P x y" shows "fmdom' x = fmdom' y"
minus_diff_sym_hmset: fixes m :: "hmultiset"   and n :: "hmultiset" shows "m - (m - n) = n - (n - m)"
Cauchy_theorem_null_homotopic: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "real \<Rightarrow> complex"   and a :: "complex" assumes "f holomorphic_on S"   and "open S"   and "valid_path g"   and "homotopic_loops S g (linepath a a)" shows "(f has_contour_integral 0) g"
fmfilter_cong: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f"   and P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> P x = Q x" shows "fmfilter P m = fmfilter Q m"
ide_char: fixes f :: "J.Arr" shows "J.ide f = (f = J.AA \<or> f = J.BB \<or> f = J.TT)"
cast_shadow_root_ptr_not_node_ptr_2: fixes node_ptr :: "(_) node_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr" shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^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 shadow_root_ptr"
Compl_lessThan: fixes k :: "'b" shows "- {..<k} = {k..}"
perp_in_comm: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "X PerpAt B A D C "
inverse_unique: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" shows "C.inv f = g"
sat: fixes cnf :: "lit uprod set" assumes "construct_sa cnf" shows "satisfiable cnf"
length_filter_le: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "length (filter P xs) \<le> length xs"
mor_T_final: shows "mor carT1 carT2 strT1 strT2 car_final1 car_final2 str_final1 str_final2 (Equiv_Relations.proj (lsbis1 carT1 carT2 strT1 strT2)) (Equiv_Relations.proj (lsbis2 carT1 carT2 strT1 strT2))"