hcju/isabelleps huggingface dataset:
tendsto_max: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and X :: "'c \<Rightarrow> 'b"   and x :: "'b"   and net :: "'c filter"   and Y :: "'c \<Rightarrow> 'b"   and y :: "'b" assumes "class.linorder_topology less_eq less open"   and "filterlim X (topological_space.nhds open x) net"   and "filterlim Y (topological_space.nhds open y) net" shows "LIM x net. ord.max less_eq (X x) (Y x) :> topological_space.nhds open (ord.max less_eq x y)"
rev_path_eq: fixes p :: "('b \<times> 'b) list"   and q :: "('b \<times> 'b) list" shows "(rev_path p = rev_path q) = (p = q)"
BRParCases': fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q \<parallel> !Q) \<in> bangRel Rel" obtains R :: "pi" and T :: "pi" and P :: "pi" and Q :: "pi" where "Pa__ = R \<parallel> P" and "Qa__ \<parallel> !Qa__ = T \<parallel> Q" and "(R, T) \<in> Rel" and "(P, Q) \<in> bangRel Rel"
canonical_proj_vimage_in_carrier: fixes I :: "'a set"   and J :: "'a set set" assumes "ideal I R"   and "J \<subseteq> carrier (R Quot I)" shows "\<Union> J \<subseteq> carrier R"
fresh_star_Un_elim: fixes S :: "'a set"   and T :: "'a set"   and c :: "'b"   and C :: "prop" shows "((S \<union> T) \<sharp>* c \<Longrightarrow> PROP C) \<equiv> (\<lbrakk>S \<sharp>* c; T \<sharp>* c\<rbrakk> \<Longrightarrow> PROP C)"
prime_not_square: fixes p :: "nat"   and k :: "nat"   and m :: "nat" assumes "prime p"   and "0 < k" shows "m * m \<noteq> p * (k * k)"
axioms_2: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R" shows "ring R"
progress_And: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> (MFOTL.And \<phi> \<psi>) j = min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)"
induct: fixes P :: "bool list \<Rightarrow> trie \<Rightarrow> bool"   and a0 :: "bool list"   and a1 :: "trie" assumes "P [] Lf"   and "\<And>b lr. P [] (Nd b lr)"   and "\<And>k ks. (\<And>x. P ks x) \<Longrightarrow> P (k # ks) Lf"   and "\<And>k ks b lr. (\<And>x. P ks x) \<Longrightarrow> P (k # ks) (Nd b lr)" shows "P a0 a1"
cos_monotone_0_pi_le: fixes y :: "real"   and x :: "real" assumes "0 \<le> y"   and "y \<le> x"   and "x \<le> pi" shows "cos x \<le> cos y"
fold_cong: fixes a :: "'e"   and b :: "'e"   and xs :: "'f list"   and ys :: "'f list"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "a = b"   and "xs = ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "fold f xs a = fold g ys b"
card_UNIV_sum: shows "CARD('a + 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 then CARD('a) + CARD('b) else 0)"
simps: fixes xs :: "'a list" shows "local.msort_key xs = local.merge_all (local.sequences xs)"
set_times_image: fixes S :: "'b set"   and T :: "'b set" shows "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)"
False: shows "\<not> (x = 0 \<or> x = 1)"
assms_1: shows "a \<noteq> a' \<or> b \<noteq> b'"
f: shows "f = (\<lambda>stp. (steps (start_of ly as + 2 * n, la, ra) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp, start_of ly as, n))"
neq_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<noteq> y) = (x < y \<or> y < x)"
checker2_eq_checker3: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "checker2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o = checker3 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o"
create_graph__name_inj: fixes \<phi> :: "'b ltlr" shows "create_graph \<phi> \<le> SPEC (inj_on name)"
mk_eqcl_fixpt_fst_nth: fixes M :: "nat bdd list \<times> bool list"   and v :: "nat"   and p :: "nat"   and q :: "nat" assumes "wf_dfa M v"   and "dfa_is_node M p"   and "dfa_is_node M q" shows "(fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! p = fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! q) = eq_nodes M v p q"
triangle_def: fixes n :: "nat" shows "triangle n = n * Suc n div 2"
least_split: fixes x :: "'c option"   and y :: "'c option"   and z :: "'c" assumes "least x y = Some z" shows "x = Some z \<or> y = Some z"
X_mult_cf: fixes p :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "p \<in> carrier P" shows "(p \<otimes>\<^bsub>P\<^esub> X_poly R) (k + 1) = p k"
hyps_1: shows "x \<sharp> \<Gamma>\<^sub>2"
fX: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real" assumes "feasible t" shows "feasible (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
all_the_same: fixes X :: "'b multiset"   and y :: "'b" assumes "\<forall>x\<in>#X. x = y" shows "card (set_mset X) \<le> Suc 0"
lemma_2_5_14: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a r\<rightarrow> b) * (b r\<rightarrow> c) \<le> a r\<rightarrow> c"
integral_mono': fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "integrable M f"   and "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"   and "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "integral\<^sup>L M g \<le> integral\<^sup>L M f"
d\<^sub>\<kappa>_proper: fixes u :: "\<nu>" shows "Semantics.d\<^sub>\<kappa> (u\<^sup>P) = Some u"
assms: shows "change_on_0 (block i t) (gap i t)"
app1: fixes l :: "nat"   and d :: "val"   and ds :: "val list"   and x :: "lambda" assumes "App l d ds \<in> calls x" shows "d \<in> vals x"
chain_homotopic_simplicial_subdivision1: fixes p :: "nat"   and q :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and r :: "nat"   and g :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "simplicial_chain p (standard_simplex q) c"   and "simplicial_simplex q (standard_simplex r) g" shows "chain_map (Suc p) g (subd p c) = subd p (chain_map p g c)"
ereal_mult_eq_PInfty: fixes a :: "ereal"   and b :: "ereal" shows "(a * b = \<infinity>) = (a = \<infinity> \<and> 0 < b \<or> 0 < a \<and> b = \<infinity> \<or> a = - \<infinity> \<and> b < 0 \<or> a < 0 \<and> b = - \<infinity>)"
DECRYPT: fixes X :: "freemsg"   and X' :: "freemsg"   and K :: "nat" assumes "X \<sim> X'" shows "DECRYPT K X \<sim> DECRYPT K X'"
le_supI2: fixes x :: "'a"   and b :: "'a"   and a :: "'a" assumes "x \<le> b" shows "x \<le> a + b"
HomFtorObj': fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "PreFunctor (HomP\<^bsub>C\<^esub>[X,\<midarrow>])"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>" shows "(HomP\<^bsub>C\<^esub>[X,\<midarrow>]) @@ Y = Hom\<^bsub>C\<^esub> X Y"
P_const_t: fixes name :: "name"   and t :: "'b" assumes "P (const name) t" shows "t = const name"
vars_eval\<^sub>B: fixes x :: "'Var"   and e :: "'BExp"   and mem :: "'Var \<Rightarrow> 'Val"   and v :: "'Val" assumes "x \<notin> bexp_vars e" shows "ev\<^sub>B mem e = ev\<^sub>B (mem(x := v)) e"
mult_strict_left_mono_neg: fixes 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 abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less b a"   and "less c zero" shows "less (times c a) (times c b)"
succ_pred: fixes i :: "'a" assumes "isSucc i" shows "succ (pred i) = i"
simps_2: fixes uv :: "real" shows "(real_min_top.R uv < PInfty) = True"
rewriteL_comp_comp: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and l :: "'c \<Rightarrow> 'd"   and h :: "'f \<Rightarrow> 'c" assumes "f \<circ> g = l" shows "f \<circ> (g \<circ> h) = l \<circ> h"
uniformly_convergent_eval_fds: fixes B :: "'a set"   and f :: "'a fds" assumes "compact B"   and "\<And>z. z \<in> B \<Longrightarrow> conv_abscissa f < ereal (z \<bullet> (1::'a))" shows "uniformly_convergent_on B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
tllist_all2_TCons2: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and ts' :: "('g, 'i) tllist"   and x :: "'h"   and ts :: "('h, 'j) tllist" shows "tllist_all2 P Q ts' (TCons x ts) = (\<exists>x' ts''. ts' = TCons x' ts'' \<and> P x' x \<and> tllist_all2 P Q ts'' ts)"
retraction_maps_eq: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c"   and f' :: "'c \<Rightarrow> 'd"   and g' :: "'d \<Rightarrow> 'c" assumes "retraction_maps X Y f g"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = f' x"   and "\<And>x. x \<in> topspace Y \<Longrightarrow> g x = g' x" shows "retraction_maps X Y f' g'"
ind: shows "\<forall>m<n. \<forall>x. m = het x \<and> 0 < m \<and> pos x \<longrightarrow> (\<exists>y. gmean x < gmean y \<and> \<gamma>_eq (x, y) \<and> het y = 0 \<and> pos y)"
PMC_SC_def: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "PMC_SC Cn selection true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL \<equiv> PartialMeetContraction Cn selection \<and> Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL \<and> Compact_logic Cn"
arr_dom: fixes f :: "nat" assumes "J'J.A.arr f" shows "J'J.A.arr (J'J.A.dom f)"
bot_ennreal_def: shows "bot \<equiv> e2ennreal 0"
order_iff_strict: fixes a :: "'a list"   and b :: "'a list" shows "prefix a b = (strict_prefix a b \<or> a = b)"
outstanding_vol_write_takeWhile_append: fixes sb :: "'b memref list"   and xs :: "'b memref list" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}" shows "takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ xs) = takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb"
P_mod_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and P :: "'e set"   and p :: "'e" shows "P_mod R S X P p = (p = \<zero>\<^bsub>R\<^esub> \<or> (\<forall>j\<le>fst (s_cf R S X p). snd (s_cf R S X p) j \<in> P))"
upper_bound_on_dom_def: fixes A :: "('a, 'b) fmap" shows "upper_bound_on_dom A = SPEC (\<lambda>n. \<forall>i\<in>#dom_m A. i < n)"
functor_axioms: shows "functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>A) local.inv"
wfP_induct_rule: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "wfP r"   and "\<And>x. (\<And>y. r y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
monic_def: fixes p :: "poly" shows "monic p = (let h = headconst p in if h = 0\<^sub>N then (p, False) else (C (Ninv h) *\<^sub>p p, 0>\<^sub>N h))"
compiler_env_config_rec0_def: fixes f :: "bool \<Rightarrow> (string \<times> string list \<times> string) option \<Rightarrow> internal_oids \<Rightarrow> nat \<times> nat \<Rightarrow> generation_semantics_toy \<Rightarrow> toy_class option \<Rightarrow> all_meta_embedding list \<Rightarrow> (string\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<times> toy_instance_single \<times> internal_oids) list \<Rightarrow> (string\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<times> (internal_oids \<times> (string \<times> toy_instance_single) toy_def_state_core) list) list \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> string\<^sub>b\<^sub>a\<^sub>s\<^sub>e list \<times> string\<^sub>b\<^sub>a\<^sub>s\<^sub>e list \<Rightarrow> string\<^sub>b\<^sub>a\<^sub>s\<^sub>e list \<Rightarrow> generation_lemma_mode option \<times> bool \<Rightarrow> 'a"   and env :: "'b compiler_env_config_scheme" shows "compiler_env_config_rec0 f env = f (D_output_disable_thy env) (D_output_header_thy env) (D_toy_oid_start env) (D_output_position env) (D_toy_semantics env) (D_input_class env) (D_input_meta env) (D_input_instance env) (D_input_state env) (D_output_header_force env) (D_output_auto_bootstrap env) (D_toy_accessor env) (D_toy_HO_type env) (D_output_sorry_dirty env)"
edom_def: fixes m :: "'b \<Rightarrow> 'c" shows "edom m = {x. m x \<noteq> \<bottom>}"
negatively_invariant_def: fixes M :: "'a set" shows "negatively_invariant M = (\<forall>x\<in>M. trapped_backward x M)"
g3cp_upRules: shows "g3cp \<subseteq> upRules"
is_reducible_lit_def: fixes Ds :: "'a literal list list"   and C :: "'a literal list"   and L :: "'a literal" shows "is_reducible_lit Ds C L = (\<exists>D\<in>set Ds. \<exists>L'\<in>set D. \<exists>\<sigma>. - L = L' \<cdot>l \<sigma> \<and> mset (remove1 L' D) \<cdot> \<sigma> \<subseteq># mset C)"
prems_4: shows "\<forall>c\<in>set (c # cs). v c \<le> n"
multlist_listassoc_cong: fixes fs :: "'a list"   and fs' :: "'a list" assumes "fs [\<sim>] fs'"   and "set fs \<subseteq> carrier G"   and "set fs' \<subseteq> carrier G" shows "foldr (\<otimes>) fs \<one> \<sim> foldr (\<otimes>) fs' \<one>"
disjI1: fixes P :: "bool"   and Q :: "bool" assumes "P" shows "P \<or> Q"
substAll_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g" shows "uninst.substAll u p g \<equiv> while (uninst.cond u p g) (uninst.step u p g) (u g, p g)"
mk_BaseDim_def: fixes d :: "'a" shows "mk_BaseDim d = dim_lambda (\<lambda>i. if i = d then 1 else 0)"
image_mset_cong2: fixes M :: "'c multiset"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and N :: "'c multiset" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x"   and "M = N" shows "image_mset f M = image_mset g N"
ideal_G: shows "ideal (extended_ord.punit.reduced_GB (homogenize None ` extend_indets ` F)) = ideal (homogenize None ` extend_indets ` F)"
sup_unique: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "\<And>x y. x \<le> f x y"   and "\<And>x y. y \<le> f x y"   and "\<And>x y z. \<lbrakk>y \<le> x; z \<le> x\<rbrakk> \<Longrightarrow> f y z \<le> x" shows "x + y = f x y"
assoc_scalar_prod: fixes v\<^sub>1 :: "'a vec"   and nr :: "nat"   and A :: "'a mat"   and nc :: "nat"   and v\<^sub>2 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec nr"   and "A \<in> carrier_mat nr nc"   and "v\<^sub>2 \<in> carrier_vec nc" shows "vec nc (\<lambda>j. v\<^sub>1 \<bullet> col A j) \<bullet> v\<^sub>2 = v\<^sub>1 \<bullet> vec nr (\<lambda>i. row A i \<bullet> v\<^sub>2)"
affine_hull_substd_basis: fixes d :: "'a set" assumes "d \<subseteq> Basis" shows "affine hull insert (0::'a) d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
assms: shows "x \<in>\<^sub>\<circ> r `\<^sub>\<circ> A"
prod_ord_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'x"   and ys :: "'y" shows "prod_ord f xs ys = prod_ord_raw f (list_of_oalistx xs) (list_of_oalisty ys)"
powr_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and A :: "(real \<times> real pdevs) \<times> real" shows "powr_aform_err p X A = (if 0 < Inf_aform_err p X then ln_aform_err p X \<bind> (\<lambda>L. exp_aform_err p (mult_aform' p A L)) else approx_bin p (powr_float_interval p) X A)"
N_below_inf_one_pp: fixes x :: "'a" shows "N x \<le> - x '"
foundation2: fixes \<tau> :: "'a state \<times> 'a state" shows "\<tau> |\<noteq> false"
wt_blocks: fixes Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "length Vs = length Ts"   and "length vs = length Ts" shows "(P,E,h \<turnstile> blocks (Vs, Ts, vs, e) : T) = (P,E(Vs [\<mapsto>] Ts),h \<turnstile> e : T \<and> (\<exists>Ts'. map typeof\<^bsub>h\<^esub> vs = map Some Ts' \<and> P \<turnstile> Ts' [\<le>] Ts))"
inv_equality: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. g (f x) = x"   and "\<And>y. f (g y) = y" shows "inv f = g"
\<Gamma>_of_mds_def: fixes mds :: "Mode \<Rightarrow> Eg1.var set" shows "\<Gamma>_of_mds mds \<equiv> \<lambda>x. if x \<notin> \<C> \<and> x \<in> mds AsmNoWrite \<union> mds AsmNoReadOrWrite then if x \<in> mds AsmNoReadOrWrite then Some {FF} else Some (dma_type x) else None"
is_closed_decr_unchanged: fixes t :: "Core.term" assumes "is_closed t" shows "decr 0 t = t"
assms: shows "h = \<^bold>g [^] x1 \<otimes> g' [^] x2"
clock_coEC_relation_image: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agents :: "'b odlist"   and s :: "'e"   and Y :: "'e odlist"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'f"   and a :: "'b"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool" assumes "FiniteLinorderEnvironment jkbp agents"   and "s \<in> toSet Y" shows "toSet (clock_knowledge envObs Y a s) = relations (mkKripke (toSet Y) (FiniteLinorderEnvironment.clock_repRels envObs) envVal) a `` {s}"
subtype_entailment: fixes t :: "'BExp set"   and P :: "'BExp set"   and t' :: "'BExp set"   and P' :: "'BExp set" assumes "t \<le>:\<^sub>P t'"   and "P' \<turnstile> P" shows "t \<le>:\<^sub>P' t'"
cases: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and a :: "('a \<times> 'b set) stream" assumes "Kripke S s0 \<delta> AP L"   and "Kripke.wfp S \<delta> AP L AP' a" obtains s :: "'a" and A :: "'b set" and \<pi> :: "('a \<times> 'b set) stream" where "a = (s, A) ## \<pi>" and "s \<in> S" and "A \<subseteq> AP'" and "A \<inter> AP = L s" and "stateOf \<pi> \<in> \<delta> s" and "Kripke.wfp S \<delta> AP L AP' \<pi>"
assms: shows "extend S ( \<LM> At i xs  \<RM> \<Rightarrow>* \<LM> At i xs  \<RM>) = ( \<Gamma> \<Rightarrow>* \<Delta>)"
axioms_2: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and basis1 :: "'d set" assumes "finite_dimensional_vector_space_pair_1_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 basis1" shows "vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
system_automaton_alt_def: shows "system_automaton = stutter_extend system_automaton'"
map_id: fixes t :: "'c option" shows "map_option id t = t"
nonneg: fixes i :: "'n"   and j :: "'n" shows "(0::'a) \<le> A $h i $h j"
ordLeq_card_Bpow: fixes r :: "('a \<times> 'a) set"   and A :: "'b set" assumes "Card_order r"   and "Field r \<noteq> {}" shows "|A| \<le>o |Bpow r A|"
renaming_preserves_cardinality: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'a set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "card (propositional_atoms.rename_clause A C) = card C"
simps_7: fixes v :: "linear_poly"   and va :: "rat" shows "nonstrict_constr (LT v va) = False"
extends_is_tree: fixes g :: "('d, 'e) rgraph_scheme"   and e :: "'d edge"   and g' :: "('d, 'e) rgraph_scheme" assumes "is_tree g"   and "ArcExt.extends g e g'" shows "is_tree g'"
frameImpResComm: fixes x :: "name"   and y :: "name"   and F :: "'b frame" shows "\<lparr>\<nu>x\<rparr>\<lparr>\<nu>y\<rparr>F \<hookrightarrow>\<^sub>F \<lparr>\<nu>y\<rparr>\<lparr>\<nu>x\<rparr>F"
h_minus_le_h_plus: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat"   and e :: "'a \<times> 'a" assumes "flow_network \<Delta> g" shows "flow_network.h_minus \<Delta> g i e \<le> flow_network.h_plus \<Delta> g i e"
comparator2_def: fixes comp1 :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and comp2 :: "'a \<Rightarrow> 'a \<Rightarrow> order" shows "comparator2 comp1 comp2 \<equiv> comparator comp1 \<and> comparator comp2"
ordinal_utility_left: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "ordinal_utility carrier relation u"   and "x \<succeq>[relation] y" shows "u y \<le> u x"
continuous_map_product_coordinates: fixes i :: "'a"   and I :: "'a set"   and T :: "'a \<Rightarrow> 'b topology" assumes "i \<in> I" shows "continuous_map (product_topology T I) (T i) (\<lambda>x. x i)"
distinct_267: fixes x12 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Prj2 x12 \<noteq> Unroll x14"
alive_update_invariant: fixes x :: "Value"   and s :: "Store"   and l :: "Location"   and y :: "Value" shows "alive x (s\<langle>l := y\<rangle>) = alive x s"
get_shadow_root_safe_locs_def: shows "get_shadow_root_safe_locs \<equiv> l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_shadow_root_safe_locs get_shadow_root_locs get_mode_locs"
replace_basis: fixes vs :: "'v list"   and us :: "'v list" assumes "set vs \<subseteq> V"   and "R_lin_independent vs"   and "length us \<le> length vs"   and "set us \<subseteq> RSpan vs"   and "R_lin_independent us" shows "\<exists>pvs. length pvs = length vs \<and> set pvs = set vs \<and> set (take (length vs) (us @ pvs)) \<subseteq> RSpan vs \<and> RSpan vs = RSpan (take (length vs) (us @ pvs)) \<and> R_lin_independent (take (length vs) (us @ pvs))"
Gamma_plus1: fixes z :: "'a" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "Gamma (z + (1::'a)) = z * Gamma z"
joins_in_V: fixes a :: "'a"   and b :: "'a"   and e :: "'edge" assumes "joins a b e"   and "e \<in> E" shows "a \<in> V \<and> b \<in> V"
finite_implies_discrete: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "finite (f ` S)" shows "\<forall>x\<in>S. \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
actionPar1Dest': fixes \<alpha> :: "'a action"   and P :: "('a, 'b, 'c) psi"   and \<beta> :: "'a action"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<alpha> \<prec> P = \<beta> \<prec> Q \<parallel> R"   and "bn \<alpha> \<sharp>* R"   and "bn \<beta> \<sharp>* R" obtains T :: "('a, 'b, 'c) psi" where "P = T \<parallel> R" and "\<alpha> \<prec> T = \<beta> \<prec> Q"
logic: fixes A :: "bool"   and B :: "bool" assumes "A \<longrightarrow> B"   and "\<not> A \<longrightarrow> B" shows "B"
smult_in_span: fixes W :: "rat vec set"   and x :: "rat vec"   and c :: "rat" assumes "W \<subseteq> Rn"   and "x \<in> gs.span W" shows "c \<cdot>\<^sub>v x \<in> gs.span W"
intros_2: fixes s :: "Value \<Rightarrow> CValue \<Rightarrow> bool"   and b :: "bool discr" shows "similar'_step s (B\<cdot>b) (CB\<cdot>b)"
finite_measure_eq_AE: fixes A :: "'a set"   and B :: "'a set" assumes "AE x in M. (x \<in> A) = (x \<in> B)"   and "A \<in> events"   and "B \<in> events" shows "prob A = prob B"
ord_isom_func: fixes E :: "('c, 'd) Order_scheme"   and f :: "'a \<Rightarrow> 'c" assumes "Order E"   and "ord_isom D E f" shows "f \<in> carrier D \<rightarrow> carrier E"
Gr_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "Gr A f = {(a, f a) |a. a \<in> A}"
e0: fixes s :: "'a" assumes "s \<in> ball x e0" shows "f' (s, u s) o\<^sub>L embed2_blinfun \<in> invertibles_blinfun"
higher_pderiv_sum: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'a poly"   and A :: "'b set" shows "(pderiv ^^ n) (sum f A) = (\<Sum>x\<in>A. (pderiv ^^ n) (f x))"