hcju/isabelleps huggingface dataset:
assms_4: shows "h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host"
iUntil_False: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "\<not> (P t'. t' \<U> t I. False)"
B_eq_Gauss: shows "B = Gauss_Jordan_in_ij A ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows)) (mod_type_class.from_nat k)"
construct_def: fixes B :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and v :: "'b" shows "construct B g v = (\<Sum>b | vs1.representation (vs1.extend_basis B) v b \<noteq> (0::'a). vs1.representation (vs1.extend_basis B) v b *b (if b \<in> B then g b else (0::'c)))"
sigma_over_set_partition_count: fixes A :: "'a set"   and P :: "'a set set"   and x :: "'a" assumes "finite A"   and "partition_on A P"   and "x \<in># \<Sum>\<^sub># (mset_set (mset_set ` P))" shows "count (\<Sum>\<^sub># (mset_set (mset_set ` P))) x = 1"
cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_inv: fixes document :: "('a, 'b, 'c, 'd option) RDocument_scheme" shows "cast (cast document) = Some document"
assms_4: fixes loc :: "'loc"   and t :: "'t" assumes "P c0"   and "next_propagate' c0 c1 loc t" shows "P c1"
BitM_plus_one: fixes n :: "Num.num" shows "Num.BitM n + num.One = num.Bit0 n"
prod_fun_imageE: fixes c :: "'d \<times> 'e"   and f :: "'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'e"   and R :: "('f \<times> 'g) set" assumes "c \<in> map_prod f g ` R" obtains x :: "'f" and y :: "'g" where "c = (f x, g y)" and "(x, y) \<in> R"
L_While_vars: fixes b :: "bexp"   and c :: "com"   and X :: "char list set" shows "vars b \<subseteq> L (WHILE b DO c) X"
h2: shows "f \<circ>\<^sub>K g \<le> g"
axioms_2: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "valid_ownership \<S> ts" shows "outstanding_volatile_writes_unowned_by_others ts"
bothstutinvs_53: fixes X :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "STUTINV X" shows "NSTUTINV f<X>"
assms_5: shows "{u..v} \<subseteq> \<Union> (U ` I)"
linear_order_strict_order: fixes x :: "'a" assumes "linear_orderable_1 x" shows "linear_orderable_2 (x \<sqinter> - (1::'a))"
tr: shows "getRel S_Top G = {(x, y). x \<in> vertices G \<and> y \<in> vertices G}"
simps_1: fixes x :: "'c list"   and ya :: "'c list" shows "(IArray x = IArray ya) = (x = ya)"
soundness: fixes H :: "'b pl set"   and p :: "'b pl" assumes "H |- p" shows "H |= p"
eSwap_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" shows "eSwap MOD zs z1 z2 ERR = ERR"
arrow_of_tabulations_in_maps_axioms: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 \<mu>"
\<V>: shows "(\<V>, \<V>') \<in> \<langle>var_rel\<rangle>set_rel"
inverse_negative_imp_negative: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less (inverse a) zero"   and "a \<noteq> zero" shows "less a zero"
subset_imageE_inj: fixes B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "B \<subseteq> f ` A" obtains C :: "'d set" where "C \<subseteq> A" and "B = f ` C" and "inj_on f C"
f_ind: fixes anything :: "nat list" shows "{\<lambda>nl. nl = xs @ 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (fft - Suc far) @ anything}"
\<tau>inv_locale: fixes t :: "'t" shows "\<tau>inv (r1 t) (r2 t) (bisim t) ta_bisim_bisim_syntax \<tau>move1 \<tau>move2"
actPres: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and \<alpha> :: "act" assumes "(P, Q) \<in> Rel" shows "\<alpha>.P \<leadsto>[Rel] \<alpha>.Q"
cases: fixes x :: "'b equation" obtains t :: "'b trm" and s :: "'b trm" where "x = Eq t s"
splitset: shows "{0..<Suc k} \<union> {Suc k} = {0..<Suc (Suc k)}"
in_arcs_in_arcs: fixes x :: "'e"   and G :: "('f, 'e) pre_digraph"   and u :: "'f" assumes "x \<in> in_arcs G u" shows "x \<in> arcs G"
dcharacter_inv_character: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "dcharacter n \<chi>" shows "dcharacter n (inv_character \<chi>)"
poly_add_length_eq: fixes K :: "'a list set"   and p1 :: "'a list list"   and p2 :: "'a list list" assumes "subring K (poly_ring R)"   and "polynomial\<^bsub>poly_ring R\<^esub> K p1"   and "polynomial\<^bsub>poly_ring R\<^esub> K p2"   and "length p1 \<noteq> length p2" shows "length (UP.poly_add p1 p2) = max (length p1) (length p2)"
pdevs_of_ivl_def: fixes l :: "'b"   and u :: "'b" shows "pdevs_of_ivl l u = scaleR_pdevs (1 / 2) (inner_scaleR_pdevs (u - l) One_pdevs)"
primepow_even_imp_primepow: fixes n :: "nat" assumes "primepow_even n" shows "primepow n"
Mp_0_smult_sdiv_poly: fixes f :: "int poly" assumes "Mp f = 0" shows "Polynomial.smult m (sdiv_poly f m) = f"
sel_3: fixes x1 :: "(String.literal \<times> typ) list"   and x2 :: "(String.literal \<times> nat) list"   and x3 :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" shows "exesorts (ExeSignature x1 x2 x3) = x3"
bell01_def: shows "|\<beta>\<^sub>0\<^sub>1\<rangle> \<equiv> complex_of_real (1 / sqrt 2) \<cdot>\<^sub>m |Matrix.vec 4 (\<lambda>i. if i = 1 \<or> i = 2 then 1 else 0)\<rangle>"
p_valid_RmiI: fixes na :: "nat"   and b :: "nat ifex"   and bdd :: "bdd" assumes "(Suc (Suc na), b) \<in> Rmi bdd" shows "pointermap_p_valid na (dpm bdd)"
ht_distinct_update: fixes l :: "('a \<times> 'b) list list"   and k :: "'a"   and v :: "'b" assumes "ht_distinct l" shows "ht_distinct (abs_update k v l)"
replacefacesAt2_nth1: fixes k :: "nat"   and ns :: "nat list"   and oldf :: "face"   and newfs :: "face list"   and F :: "face list list" assumes "k \<notin> set ns" shows "replacefacesAt2 ns oldf newfs F ! k = F ! k"
spr_sim_simps: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agent :: "'b"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and t :: "'e Trace" assumes "FiniteSingleAgentEnvironment jkbp agent" shows "fst (FiniteSingleAgentEnvironment.spr_sim jkbp envInit envAction envTrans envVal envObs agent t) = FiniteSingleAgentEnvironment.spr_abs jkbp envInit envAction envTrans envVal envObs agent t"
proj_lift_poly_fun_homogenous: fixes f :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and k :: "'k"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "f 0 = 0" shows "proj_poly k (lift_poly_fun f p) = f (proj_poly k p)"
gtrace_infinite: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and r :: "(nat \<times> 'd) stream"   and v :: "nat \<times> 'd" assumes "grun A w V r v" shows "infinite (sset (gtrace r v))"
approx_\<beta>_closure_\<alpha>': fixes Z :: "('c \<Rightarrow> real) set" assumes "Z \<in> V'" shows "Approx\<^sub>\<beta> Z \<subseteq> Closure\<^sub>\<alpha> Z"
rtrancl_reachable_induct: fixes I :: "'a set"   and INV :: "'a set"   and E :: "('a \<times> 'a) set" assumes "I \<subseteq> INV"   and "E `` INV \<subseteq> INV" shows "E\<^sup>* `` I \<subseteq> INV"
wt_elim_cases_27: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and x :: "tys" assumes "E,dt\<Turnstile>In1r (Throw e)\<Colon>x" obtains tn :: "qtname" where "x = Inl (PrimT Void)" and "E,dt\<Turnstile>e\<Colon>-Class tn" and "prg E\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable"
dfs_invariant: fixes S :: "nat option bdd \<times> bool list list"   and xs :: "bool list list"   and I :: "nat option bdd \<times> bool list list \<Rightarrow> bool" assumes "subset_invariant A S"   and "list_all (nfa_is_node A) xs"   and "I S"   and "\<And>S x. \<lbrakk>\<not> subset_memb x S; nfa_is_node A x; subset_invariant A S; I S\<rbrakk> \<Longrightarrow> I (subset_ins x S)" shows "I (dfs S xs)"
prems: shows "\<forall>x\<in>set (x # xs). ChamberComplexEndomorphism X (f x)"
pi: shows "set pi \<subseteq> Xs \<times> Ys"
test_mult: fixes x :: "'a"   and y :: "'a" shows "n n (n x \<cdot> n y) = n x \<cdot> n y"
induct: fixes x1 :: "nat"   and x2 :: "nat"   and x3 :: "((nat \<times> (form, nat) out_port) \<times> nat \<times> (form, nat) antecedent) list"   and P :: "nat \<Rightarrow> nat \<Rightarrow> ((nat \<times> (form, nat) out_port) \<times> nat \<times> (form, nat) antecedent) list \<Rightarrow> bool" assumes "task.path x1 x2 x3"   and "\<And>v. P v v []"   and "\<And>e v' pth. \<lbrakk>e \<in> task_edges; task.path (task.edge_end e) v' pth; P (task.edge_end e) v' pth\<rbrakk> \<Longrightarrow> P (task.edge_begin e) v' (e # pth)" shows "P x1 x2 x3"
conjugate_exponent_ennreal_3: fixes p :: "ennreal" assumes "1 \<le> p" shows "conjugate_exponent (conjugate_exponent p) = p"
readable: shows "\<forall>asset\<in>stocks Mkt. borel_adapt_stoch_proc F (prices Mkt asset)"
Subset_Zero_Iff: shows "{} \<turnstile> Var i SUBS Zero IFF All2 j (Var i) Fls"
finite_SetCompr2: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b" assumes "finite (Collect P)"   and "\<forall>y. P y \<longrightarrow> finite (range (f y))" shows "finite {f y x |x y. P y}"
state_to_strips_state_effect_consistent: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and v :: "'c"   and a :: "'d"   and s :: "'c \<Rightarrow> 'd option"   and a' :: "'d" assumes "is_valid_problem_sas_plus \<Psi>"   and "(v, a) \<in> dom (\<phi>\<^sub>S \<Psi> s)"   and "(v, a') \<in> dom (\<phi>\<^sub>S \<Psi> s)"   and "(\<phi>\<^sub>S \<Psi> s) (v, a) = Some True"   and "(\<phi>\<^sub>S \<Psi> s) (v, a') = Some True" shows "(v, a) = (v, a')"
finite_conv_nat_seg_image: fixes A :: "'b set" shows "finite A = (\<exists>n f. A = f ` {i. i < n})"
set_cases_1: fixes e :: "'c"   and a :: "('c, 'd) blindable\<^sub>m" assumes "e \<in> set1_blindable\<^sub>m a" obtains "a = Unblinded e"
lipschitz_on_uniformly_continuous: fixes L :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "L-lipschitz_on X f" shows "uniformly_continuous_on X f"
gb_schema_dummy_eq_gb_schema_aux: fixes data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "struct_spec sel ap ab compl" shows "fst (gb_schema_dummy data D bs ps) = gb_schema_aux data bs ps"
v_2: shows "v\<^sub>2 \<in> carrier_vec n"
elims: fixes x :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and y :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "sig x = y" obtains \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" and uu_ :: "Core.term set" where "x = (\<Sigma>, uu_)" and "y = \<Sigma>"
induct: fixes x1 :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and x2 :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and P :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set \<Rightarrow> 'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set \<Rightarrow> bool" assumes "x1 \<leadsto>RP x2"   and "\<And>A C N P Q. \<lbrakk>Neg A \<in># C; Pos A \<in># C\<rbrakk> \<Longrightarrow> P (N \<union> {C}, P, Q) (N, P, Q)"   and "\<And>D P Q C N. \<lbrakk>D \<in> P \<union> Q; subsumes D C\<rbrakk> \<Longrightarrow> P (N \<union> {C}, P, Q) (N, P, Q)"   and "\<And>D N C P Q. \<lbrakk>D \<in> N; strictly_subsumes D C\<rbrakk> \<Longrightarrow> P (N, P \<union> {C}, Q) (N, P, Q)"   and "\<And>D N C P Q. \<lbrakk>D \<in> N; strictly_subsumes D C\<rbrakk> \<Longrightarrow> P (N, P, Q \<union> {C}) (N, P, Q)"   and "\<And>D L' P Q L \<sigma> C N. \<lbrakk>D + {#L'#} \<in> P \<union> Q; - L = L' \<cdot>l \<sigma>; D \<cdot> \<sigma> \<subseteq># C\<rbrakk> \<Longrightarrow> P (N \<union> {C + {#L#}}, P, Q) (N \<union> {C}, P, Q)"   and "\<And>D L' N L \<sigma> C P Q. \<lbrakk>D + {#L'#} \<in> N; - L = L' \<cdot>l \<sigma>; D \<cdot> \<sigma> \<subseteq># C\<rbrakk> \<Longrightarrow> P (N, P \<union> {C + {#L#}}, Q) (N, P \<union> {C}, Q)"   and "\<And>D L' N L \<sigma> C P Q. \<lbrakk>D + {#L'#} \<in> N; - L = L' \<cdot>l \<sigma>; D \<cdot> \<sigma> \<subseteq># C\<rbrakk> \<Longrightarrow> P (N, P, Q \<union> {C + {#L#}}) (N, P \<union> {C}, Q)"   and "\<And>N C P Q. P (N \<union> {C}, P, Q) (N, P \<union> {C}, Q)"   and "\<And>N Q C P. N = old_concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) Q C) \<Longrightarrow> P ({}, P \<union> {C}, Q) (N, P, Q \<union> {C})" shows "P x1 x2"
inj_\<iota>: shows "inj_on \<iota> U"
assms_4: shows "recfn (Suc n) f\<^sub>1"
args_subst_db: fixes i :: "nat"   and x :: "'a"   and s :: "('s, 'a) tm" shows "args (subst_db i x s) = map (subst_db (if head s = Sym lam then i + 1 else i) x) (args s)"
1: fixes u :: "'n" assumes "u \<in> B" shows "norm u = 1"
\<rho>': fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and d :: "int" assumes "iszlfm p (a # bs)"   and "d_\<delta> p d"   and "0 < d" shows "\<forall>x. \<not> (\<exists>(e, c)\<in>set (\<rho> p). \<exists>j\<in>{1..c * d}. Ifm (a # bs) (\<sigma> p c (Add e (C j)))) \<longrightarrow> Ifm (real_of_int x # bs) p \<longrightarrow> Ifm (real_of_int (x - d) # bs) p"
ta_hb_consistent_coinduct_append_wf: fixes X :: "('a \<times> ('b, 'a) obs_event action) list \<Rightarrow> ('a \<times> ('b, 'a) obs_event action) llist \<Rightarrow> 'c \<Rightarrow> bool"   and E :: "('a \<times> ('b, 'a) obs_event action) list"   and obs :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "'c"   and R :: "('c \<times> 'c) set"   and P :: "'d prog" assumes "X E obs a"   and "wf R"   and "\<And>E obs a. X E obs a \<Longrightarrow> obs = LNil \<or> (\<exists>obs' obs'' a'. obs = lappend obs' obs'' \<and> ta_hb_consistent P E obs' \<and> (obs' = LNil \<longrightarrow> (a', a) \<in> R) \<and> (lfinite obs' \<longrightarrow> X (E @ list_of obs') obs'' a' \<or> ta_hb_consistent P (E @ list_of obs') obs''))" shows "ta_hb_consistent P E obs"
infinite_times_eqpoll_self: fixes A :: "'b set" assumes "infinite A" shows "A \<times> A \<approx> A"
li_smaller_than_gen: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "A \<subseteq> carrier V"   and "B \<subseteq> carrier V"   and "V.module.lin_indpt A"   and "V.module.gen_set B" shows "card A \<le> card B"
iT_Div_icard_ge: fixes I :: "nat set"   and d :: "nat" shows "icard I div enat d + enat (if icard I mod enat d = 0 then 0 else Suc 0) \<le> icard (I \<oslash> d)"
iT_mult_2: fixes n :: "nat"   and k :: "nat" shows "[n\<dots>] \<otimes> k = [ n * k, mod k ]"
elims: fixes x :: "eint"   and y :: "int" assumes "fromeint x = y" obtains xa :: "int" where "x = eint xa" and "y = xa"   | "x = \<infinity>" and "y = undefined"
utp_order_le: fixes T :: "'a hrel \<Rightarrow> 'a hrel" shows "le (utp_order T) = (\<sqsubseteq>)"
tendsto: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'a"   and a :: "'a"   and F :: "'d filter"   and g :: "'d \<Rightarrow> 'b"   and b :: "'b" assumes "bounded_bilinear prod"   and "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. prod (f x) (g x)) \<longlongrightarrow> prod a b) F"
derivation_ge_append: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat" shows "derivation_ge (D @ E) i = (derivation_ge D i \<and> derivation_ge E i)"
ag_mOp_closed: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c" assumes "aGroup A"   and "x \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> x \<in> carrier A"
simps: fixes curr_type :: "checker_type"   and cons :: "ct list"   and prods :: "checker_type" shows "type_update curr_type cons prods = produce (consume curr_type cons) prods"
f_1: shows "integrable (lebesgue_on {0..a}) f"
leg_simps_4: shows "C.dom trg_f.cod.leg1 = trg_f.cod.apex"
triangle': fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "f \<star> \<l>[g] = (\<r>[f] \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[f, src f, g]"
pdevs_val_add: fixes e :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real"   and xs :: "'b pdevs" shows "pdevs_val (\<lambda>i. e i + f i) xs = pdevs_val e xs + pdevs_val f xs"
sum_le: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real"   and b :: "real" assumes "finite S"   and "\<forall>r\<in>S. f r \<le> b" shows "sum f S \<le> real (card S) * b"
2: shows "step' \<top> c' \<sqsubseteq> c'"
labels_psimps_9: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and fmsg :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "labels \<Gamma> ({l}receive(fmsg) . p) = {l}"
G'_no_deadends: fixes v :: "'a" assumes "v \<in> V'" shows "\<not> \<not> (\<exists>w\<in>V\<^bsub>subgame V'\<^esub>. v \<rightarrow>\<^bsub>subgame V'\<^esub> w)"
span_add: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and S :: "'d set"   and y :: "'d" assumes "module scale"   and "x \<in> module.span scale S"   and "y \<in> module.span scale S" shows "x + y \<in> module.span scale S"
1: shows "f \<approx>>\<^bsub>CatExp' A B\<^esub> g"
sel_3: fixes x1 :: "'g option" shows "Resumption.resumption.resume (Resumption.resumption.Done x1) = (\<lambda>inp. Resumption.resumption.Done None)"
isCont_p': fixes x :: "real"   and a :: "real"   and v :: "real" shows "isCont (movement.p' a v) x"
indicator_sum_eq: fixes m :: "real"   and n :: "nat"   and f :: "'c \<Rightarrow> real"   and x :: "'c" assumes "\<bar>m\<bar> \<le> 2 ^ (2 * n)"   and "m / 2 ^ n \<le> f x"   and "f x < (m + 1) / 2 ^ n"   and "m \<in> \<int>" shows "(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2 * n). k / 2 ^ n * indicat_real {y. k / 2 ^ n \<le> f y \<and> f y < (k + 1) / 2 ^ n} x) = m / 2 ^ n"
postListpostSet: fixes xs :: "('e \<times> ('f \<Rightarrow> 'g) \<times> 'h) list"   and l :: "'e \<Rightarrow> 'g"   and s :: "'f" assumes "postSet (set xs) l s" shows "postList xs l s"
jw_Rec_inv: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and x :: "Syntax.var"   and t :: "Syntax.term"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile>\<^sub>W Rec x t : \<tau>"   and "atom x \<sharp> \<Gamma>" obtains y :: "Syntax.var" and e :: "Syntax.term" and \<tau>\<^sub>1 :: "ty" and \<tau>\<^sub>2 :: "ty" where "atom y \<sharp> (\<Gamma>, x)" and "t = Lam y e" and "\<tau> = Fun \<tau>\<^sub>1 \<tau>\<^sub>2" and "\<Gamma>(x $$:= Fun \<tau>\<^sub>1 \<tau>\<^sub>2) \<turnstile>\<^sub>W Lam y e : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
Rec_functional: fixes zero :: "'a"   and succ :: "'a \<Rightarrow> 'a"   and e :: "'b"   and r :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'a" assumes "peano zero succ" shows "\<exists>!y. peano.Rec zero succ e r x y"
x1_def: shows "x1 = Abs_real_int (left x, s)"
showsp_list_code: fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and p :: "nat"   and xs :: "'a list" shows "showsp_list s p xs = shows_list_gen (s 0) ''[]'' ''['' '', '' '']'' xs"
inter_Collect_eq_empty: fixes X0 :: "'b set"   and g :: "'b \<Rightarrow> bool" assumes "\<And>x. x \<in> X0 \<Longrightarrow> \<not> g x" shows "X0 \<inter> Collect g = {}"
cmp_transpos1: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j"   and "k \<le> n" shows "cmp (transpos i j) (transpos i j) k = k"
composed_arg_cong: fixes X :: "msg"   and Y :: "msg" assumes "X = Y" shows "composed X = composed Y"
equivalence_def: fixes S :: "('b, 'c) eq_object_scheme" shows "equivalence S \<equiv> (\<forall>x. x \<in> carrier S \<longrightarrow> x .=\<^bsub>S\<^esub> x) \<and> (\<forall>x y. x .=\<^bsub>S\<^esub> y \<longrightarrow> x \<in> carrier S \<longrightarrow> y \<in> carrier S \<longrightarrow> y .=\<^bsub>S\<^esub> x) \<and> (\<forall>x y z. x .=\<^bsub>S\<^esub> y \<longrightarrow> y .=\<^bsub>S\<^esub> z \<longrightarrow> x \<in> carrier S \<longrightarrow> y \<in> carrier S \<longrightarrow> z \<in> carrier S \<longrightarrow> x .=\<^bsub>S\<^esub> z)"
disj_lin: fixes p :: "fm"   and q :: "fm" assumes "isrlfm p"   and "isrlfm q" shows "isrlfm (Ferrack.disj p q)"
rsp: shows "\<epsilon>\<^sub>c \<simeq> \<epsilon>\<^sub>c"
test_star: fixes p :: "'a" shows "\<iota> p\<^sup>\<star> = (1::'b)"
f'_def: shows "f' = (inverse (x + a) - inverse x) / a"
intros_7: fixes CT :: "nat \<Rightarrow> classDef option"   and ei :: "exp"   and ei' :: "exp"   and C :: "nat"   and el :: "exp list"   and er :: "exp list" assumes "CT \<turnstile> ei \<rightarrow> ei'" shows "CT \<turnstile> New C (el @ ei # er) \<rightarrow> New C (el @ ei' # er)"
transitive_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "transitive x"   and "transitive y" shows "transitive (x \<sqinter> y)"
simps_1: fixes x :: "udom"   and f :: "udom \<rightarrow> udom\<cdot>'a resT" shows "bindU\<cdot>(Done\<cdot>x)\<cdot>f = f\<cdot>x"
rep_not_zero: shows "\<Lambda> * (\<v> - 1) div (\<k> - 1) \<noteq> 0"
RSpanI: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list" assumes "RModule R smult M"   and "set ms \<subseteq> M"   and "M \<subseteq> R_scalar_mult.RSpan R smult ms" shows "M = R_scalar_mult.RSpan R smult ms"
INV_leadsto: fixes I :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "\<turnstile> \<box>I \<and> (P \<and> I \<leadsto> Q) \<longrightarrow> (P \<leadsto> Q)"
infnorm_Max: fixes x :: "'a" shows "infnorm x = (MAX i\<in>Basis. \<bar>x \<bullet> i\<bar>)"
Max_in: fixes A :: "complex set" assumes "finite A"   and "A \<noteq> {}" shows "complex_lex.Max A \<in> A"
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and G :: "'b \<Rightarrow> 'a" assumes "\<exists>F. inverse_functors A B F G" shows "invertible_functor_axioms A B G"
ZFfunIdLeft: fixes f :: "ZF" assumes "isZFfun f" shows "ZFfun (|dom|f) (|dom|f) (\<lambda>x. x) |o| f = f"
shallower_pos: fixes t :: "('a, 'b) tm"   and p :: "dir list"   and q :: "dir list"   and dq :: "dir"   and dp :: "dir" assumes "position_of t (p @ q @ [dq])" shows "position_of t (p @ [dp])"
linear_injective_0: fixes f :: "'c \<Rightarrow> 'd" assumes "linear f" shows "inj f = (\<forall>x. f x = (0::'d) \<longrightarrow> x = (0::'c))"