hcju/isabelleps huggingface dataset:
simps_6: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and b2 :: "bool" shows "GExp_Lexorder.less_gexp_aux (Eq e1 e2) (Bc b2) = False"
rep_eq: fixes x :: "Float.float" shows "is_float_zero x = (0 = real_of_float x)"
order_iff_strict: fixes b :: "nat list"   and a :: "nat list" shows "(b \<le>\<^sub>v a) = (b <\<^sub>v a \<or> a = b)"
induct: fixes x :: "'b tensor"   and P :: "'b tensor \<Rightarrow> bool" assumes "cprank_max1 x"   and "\<And>A. order A \<le> 1 \<Longrightarrow> P A"   and "\<And>A B. \<lbrakk>order A = 1; cprank_max1 B; P B\<rbrakk> \<Longrightarrow> P (A \<otimes> B)" shows "P x"
distinct_2: fixes x11 :: "name"   and x12 :: "name" shows "\<tau> \<noteq> x11[x12]"
median_def: fixes xs :: "'b list" shows "median xs = select ((length xs - 1) div 2) xs"
SUP_upper: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and i :: "'h"   and A :: "'h set"   and f :: "'h \<Rightarrow> 'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
ide_map\<^sub>0_obj: fixes a :: "'a" assumes "C.obj a" shows "D.ide (\<tau>'\<tau>.map\<^sub>0 a)"
CONSTRAINT_def: fixes P :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "CONSTRAINT P x \<equiv> P x"
I_def: shows "I = (\<lambda>i. case map_of (map prod.swap pp) i of None \<Rightarrow> 0::'a | Some j \<Rightarrow> - A $$ (j, qj))"
results_conv_fixp: shows "results \<equiv> set.fixp_fun (\<lambda>results r. case r of Resumption.resumption.Done x \<Rightarrow> set_option x | Resumption.resumption.Pause out c \<Rightarrow> \<Union>input. results (c input))"
one_le_inverse_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 x :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq one (inverse x) = (less zero x \<and> less_eq x one)"
lower_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and t :: "'c" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.lower ord_term p t = except p {s. ord_term t s}"
if_split_mem2: fixes a :: "'d"   and Q :: "bool"   and x :: "'d set"   and y :: "'d set" shows "(a \<in> (if Q then x else y)) = ((Q \<longrightarrow> a \<in> x) \<and> (\<not> Q \<longrightarrow> a \<in> y))"
ag_add4_rel: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus> b \<plusminus> (c \<plusminus> d) = a \<plusminus> c \<plusminus> (b \<plusminus> d)"
eval_nembed_true: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b" assumes "P s" shows "\<guillemotleft> \<N> P \<guillemotright> s = 0"
hom_graph_hom: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a formula" shows "\<exists>S. hom_graph f S x (hom f x)"
le_filter_frequently: fixes F :: "'b filter"   and G :: "'b filter" shows "(F \<le> G) = (\<forall>P. frequently P F \<longrightarrow> frequently P G)"
vconn_implies_hpar_3: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>S g\<guillemotright>" shows "S.src g = S.src f"
boundE: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d" assumes "bound X r b" obtains "\<And>x. x \<in> X \<Longrightarrow> r x b"
list_of_lmap: fixes xs :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "lfinite xs" shows "list_of (lmap f xs) = map f (list_of xs)"
igWlsAllDisj_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igWlsAllDisj MOD \<equiv> igWlsDisj MOD \<and> igWlsAbsDisj MOD"
D: shows "set_mset K \<subseteq> r \<down>m J"
quasinorm_subsetI: fixes N1 :: "'b quasinorm"   and N2 :: "'b quasinorm"   and C :: "real" assumes "\<And>f. f \<in> space\<^sub>N N1 \<Longrightarrow> eNorm N2 f \<le> ennreal C * eNorm N1 f" shows "N1 \<subseteq>\<^sub>N N2"
subset_inj_on: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'c set" assumes "inj_on f B"   and "A \<subseteq> B" shows "inj_on f A"
quotient_of_submodules_is_ideal: fixes P :: "'a set"   and Q :: "'a set" assumes "submodule R M P"   and "submodule R M Q" shows "ideal R (P \<^bsub>R\<ddagger>M\<^esub> Q)"
echelon_form_upt_k_intro: fixes k :: "nat"   and A :: "(('e, 'f) vec, 'd) vec" assumes "\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)"   and "\<forall>i j. i < j \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k j k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'e)) < (LEAST n. A $ j $ n \<noteq> (0::'e))" shows "echelon_form_upt_k A k"
unat_sub: fixes b :: "'b word"   and a :: "'b word" assumes "b \<le> a" shows "unat (a - b) = unat a - unat b"
vproduct_vdoubletonD_2: fixes f :: "V"   and a :: "V"   and b :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct (set {a, b}) A" shows "f\<lparr>a\<rparr> \<in>\<^sub>\<circ> A a"
bl_to_bin_lt2p_drop: fixes bs :: "bool list" shows "bl_to_bin bs < 2 ^ length (dropWhile Not bs)"
assoc'_agreement: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "C.assoc' a b c = assoc' a b c"
diff_right_mono: 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq a b" shows "less_eq (minus a c) (minus b c)"
ecard2: fixes E :: "'a uprod set"   and e :: "'a uprod" assumes "uGraph E"   and "e \<in> E" shows "proper_uprod e"
diff_submanifold: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and S :: "'c set" assumes "diff k charts1 charts2 f"   and "open S" shows "diff k (manifold.charts_submanifold charts1 S) charts2 f"
negatively_invariant_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.negatively_invariant f X M = (\<forall>x\<in>M. auto_ll_on_open.trapped_backward f X x M)"
hc: shows "\<forall>v. insertion (nth_default 0 (L[var := v])) c = C"
distr_id: fixes N :: "'c measure" shows "distr N N (\<lambda>x. x) = N"
has_vderiv_on_cong: fixes S :: "real set"   and f :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and g' :: "real \<Rightarrow> 'b"   and T :: "real set" assumes "\<And>x. x \<in> S \<Longrightarrow> f x = g x"   and "\<And>x. x \<in> S \<Longrightarrow> f' x = g' x"   and "S = T" shows "(f has_vderiv_on f') S = (g has_vderiv_on g') T"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat"   and a3 :: "varDef list" assumes "fields(a1,a2) = a3" obtains CT :: "nat \<Rightarrow> classDef option" where "a1 = CT__" and "a2 = Object" and "a3 = []"   | CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and CDef :: "classDef" and D :: "nat" and Cf :: "varDef list" and Dg :: "varDef list" and DgCf :: "varDef list" where "a1 = CT" and "a2 = C" and "a3 = DgCf" and "CT C = Some CDef" and "cSuper CDef = D" and "cFields CDef = Cf" and "fields(CT,D) = Dg" and "DgCf = Dg @ Cf"
threshold_closed: fixes \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set" assumes "closed \<G> w" shows "\<exists>k. \<forall>\<psi>. G \<psi> \<in> \<G> \<longrightarrow> threshold \<psi> (suffix k w) \<G> = Some 0"
simps_5: fixes I :: "('b, 'c, 'a) interp"   and r :: "real"   and v :: "(real, 'a) vec" shows "frechet I (Const r) v = (\<lambda>v'. 0)"
stermsl_no_choice: fixes p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" shows "p1 \<oplus> p2 \<notin> stermsl p"
parallel_pair_diagram_axioms: shows "parallel_pair_diagram (\<cdot>) f0 f1"
solution_set_rel: fixes p :: "('b, 'c) vec"   and A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "is_solution p A b" shows "solution_set A b = {p} + solution_set A 0"
reachable_transitive: fixes x :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and y :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and z :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" assumes "reachable x y"   and "reachable y z" shows "reachable x z"
abs_eq: fixes xa :: "nat"   and x :: "nat \<Rightarrow> 'a" shows "Matrix.vec xa x = Abs_vec (xa, mk_vec xa x)"
h1: shows "mapping_of (a + b) = mapping_of a + mapping_of b"
inputSupportDerivative: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<x> \<prec> P'" shows "supp P' - {x} \<subseteq> supp P"
intruder_synth_timpl_closure': fixes M :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term set"   and TI :: "('sets set \<times> 'sets set) set"   and t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and s :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term" assumes "timpl_closure_set M TI \<turnstile>\<^sub>c t"   and "s \<in> timpl_closure t TI" shows "timpl_closure_set M TI \<turnstile>\<^sub>c s"
square_free_factorization_order_root_mem: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list"   and a :: "'a poly"   and i :: "nat"   and x :: "'a" assumes "square_free_factorization p (c, bs)"   and "p \<noteq> 0"   and "(a, i) \<in> set bs"   and "poly a x = (0::'a)" shows "order x p = Suc i"
TArr_def: fixes x2 :: "ty"   and x1 :: "ty" shows "x2 \<rightarrow> x1 \<equiv> Abs_ty (TArr_Rep (Rep_ty x2) (Rep_ty x1))"
second_vertices_card: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set" assumes "DisjointPaths G v0 v1 paths" shows "card (DisjointPaths.second_vertices paths) = card paths"
fusion_partially_correct: fixes wrap :: "'b \<rightarrow> 'a"   and unwrap :: "'a \<rightarrow> 'b"   and work :: "'b"   and body :: "'a \<rightarrow> 'a"   and body' :: "'b \<rightarrow> 'b" assumes "wrap oo unwrap = ID"   and "work = fix\<cdot>(unwrap oo body oo wrap)"   and "unwrap oo body oo wrap = body' oo unwrap oo wrap" shows "fix\<cdot>body' \<sqsubseteq> work"
sum_trading_strat: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "trading_strategy pf1"   and "trading_strategy pf2" shows "trading_strategy (qty_sum pf1 pf2)"
invar_trie_delete: fixes t :: "('c, 'd) Trie.trie"   and ks :: "'c list" assumes "invar_trie t" shows "invar_trie (delete_trie ks t)"
prems_2: shows "f \<in> insert x F \<rightarrow> carrier G"
exec_call_Normal_elim: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> \<Rightarrow> t" obtains bdy :: "('e, 'd, 'f) com" and t' :: "'e" where "\<Gamma> p = Some bdy__" and "\<Gamma>\<turnstile> \<langle>bdy__,Normal (init s)\<rangle> \<Rightarrow> Normal t'__" and "\<Gamma>\<turnstile> \<langle>c s t'__,Normal (return s t'__)\<rangle> \<Rightarrow> t"   | bdy :: "('e, 'd, 'f) com" and t' :: "'e" where "\<Gamma> p = Some bdya__" and "\<Gamma>\<turnstile> \<langle>bdya__,Normal (init s)\<rangle> \<Rightarrow> Abrupt t'" and "t = Abrupt (return s t')"   | bdy :: "('e, 'd, 'f) com" and f :: "'f" where "\<Gamma> p = Some bdyb__" and "\<Gamma>\<turnstile> \<langle>bdyb__,Normal (init s)\<rangle> \<Rightarrow> Fault f" and "t = Fault f"   | bdy :: "('e, 'd, 'f) com" where "\<Gamma> p = Some bdy" and "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Stuck" and "t = Stuck"   | "\<Gamma> p = None" and "t = Stuck"
lset_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lset (lappend xs ys) \<subseteq> lset xs \<union> lset ys"
sup_commute: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) (y::'b) = sup y x"
vbrelationD1: fixes r :: "V"   and x :: "V" assumes "vbrelation r"   and "x \<in>\<^sub>\<circ> r" shows "\<exists>a b. x = \<langle>a, b\<rangle>"
split_option_bind: fixes P :: "'c option \<Rightarrow> bool"   and res :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "P (option_bind res f) = ((res = None \<longrightarrow> P None) \<and> (\<forall>s. res = Some s \<longrightarrow> P (f s)))"
if_eval_elim': fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and e :: "'c"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and mds :: "Mode \<Rightarrow> 'b set"   and mem :: "'b \<Rightarrow> 'd"   and c' :: "('b, 'a, 'c) Stmt"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'b \<Rightarrow> 'd" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "(((Stmt.If e c\<^sub>1 c\<^sub>2, mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B" shows "(c' = c\<^sub>1 \<and> ev\<^sub>B mem e \<or> c' = c\<^sub>2 \<and> \<not> ev\<^sub>B mem e) \<and> mds' = mds \<and> mem' = mem"
ImageE: fixes b :: "'a"   and r :: "('b \<times> 'a) set"   and A :: "'b set" assumes "b \<in> r `` A" obtains x :: "'b" where "(x, b) \<in> r" and "x \<in> A"
lmap_snd_lzip_conv_ltake: fixes xs :: "'d llist"   and ys :: "'c llist" shows "lmap snd (lzip xs ys) = ltake (min (llength xs) (llength ys)) ys"
lr_theta_rep_def: shows "lr_theta_rep \<equiv> \<lambda>(rm, rp). {(f, f'). \<forall>(x, x')\<in>unlr (fst (undual rm)). (f\<cdot>x, f'\<cdot>x') \<in> unlr (fst rp)}"
add: fixes f :: "'m \<Rightarrow> 'n"   and x :: "'m"   and y :: "'m" assumes "additive f" shows "f (x + y) = f x + f y"
neg_le_0_iff_le: fixes 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 a :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) zero = less_eq zero a"
simps_3: fixes v :: "nat" shows "isnpolyh (Bound v) = (\<lambda>k. False)"
VVV_ide_char: fixes \<mu>\<nu>\<tau> :: "'a \<times> 'a \<times> 'a" shows "VVV.ide \<mu>\<nu>\<tau> = B.VVV.ide (DN \<mu>\<nu>\<tau>)"
Rep_Abs_Hiding: fixes A :: "'b set"   and P :: "'b process" shows "Rep_process (Abs_process ({(s, X). \<exists>t. s = trace_hide t (ev ` A) \<and> (t, X \<union> ev ` A) \<in> F P} \<union> {(s, X). s \<in> div_hide P A}, div_hide P A)) = ({(s, X). \<exists>t. s = trace_hide t (ev ` A) \<and> (t, X \<union> ev ` A) \<in> F P} \<union> {(s, X). s \<in> div_hide P A}, div_hide P A)"
integrable_on_lborel: fixes f :: "'a \<Rightarrow> 'b" assumes "integrable lborel f" shows "f integrable_on UNIV"
perm_vars_length: fixes n :: "nat"   and vs :: "nat list" assumes "perm_vars n vs" shows "length vs = n"
subset_path_image_join: fixes g1 :: "real \<Rightarrow> 'a"   and s :: "'a set"   and g2 :: "real \<Rightarrow> 'a" assumes "path_image g1 \<subseteq> s"   and "path_image g2 \<subseteq> s" shows "path_image (g1 +++ g2) \<subseteq> s"
inject: fixes x1 :: "char list \<Rightarrow> 'b option"   and y1 :: "char list \<Rightarrow> 'b option" shows "(Env x1 = Env y1) = (x1 = y1)"
HLD_def: fixes s :: "'a set" shows "HLD s = holds (\<lambda>x. x \<in> s)"
zero_transpose_mat: fixes n :: "nat"   and m :: "nat" shows "(0\<^sub>m n m)\<^sup>T = 0\<^sub>m m n"
not_eq_order_implies_strict: fixes a :: "('d, 'e) Term.term"   and b :: "('d, 'e) Term.term" assumes "a \<noteq> b"   and "a \<unrhd> b" shows "a \<rhd> b"
a_Pair: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e\<^sub>1 :: "Syntax.term"   and eP\<^sub>1 :: "Syntax.term"   and eV\<^sub>1 :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and e\<^sub>2 :: "Syntax.term"   and eP\<^sub>2 :: "Syntax.term"   and eV\<^sub>2 :: "Syntax.term"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e\<^sub>1, eP\<^sub>1, eV\<^sub>1 : \<tau>\<^sub>1"   and "\<Gamma> \<turnstile> e\<^sub>2, eP\<^sub>2, eV\<^sub>2 : \<tau>\<^sub>2" shows "\<Gamma> \<turnstile> Syntax.Pair e\<^sub>1 e\<^sub>2, Syntax.Pair eP\<^sub>1 eP\<^sub>2, Syntax.Pair eV\<^sub>1 eV\<^sub>2 : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2"
stermsl_in_wfpf: fixes A :: "('a, 'b, 'c, 'd) seqp set"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "A \<subseteq> {(q, p). p \<leadsto>\<^bsub>\<Gamma>\<^esub> q} `` A"   and "p \<in> A" shows "\<exists>r\<in>stermsl p. r \<in> A"
fold_congs_1: fixes r :: "'b control_state_scheme"   and r' :: "'b control_state_scheme"   and v' :: "bool"   and f :: "bool \<Rightarrow> bool"   and f' :: "bool \<Rightarrow> bool" assumes "r = r'"   and "break_status r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "break_status_update f r = break_status_update f' r'"
prems_1: shows "\<delta>' = remap_rule f ` \<delta>"
y'_def: shows "y' = geodesic_segment_param {py--y} py M"
gauss_jordan_single_3: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and C :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan_single A = C" shows "row_echelon_form C"
unique_member_of: fixes G :: "prog"   and n :: "qtname \<times> memberdecl"   and C :: "qtname"   and m :: "qtname \<times> memberdecl" assumes "G \<turnstile> n member_of C"   and "G \<turnstile> m member_of C"   and "memberid n = memberid m" shows "n = m"
generate_field_is_subfield: fixes H :: "'a set" assumes "H \<subseteq> carrier R" shows "subfield (generate_field R H) R"
plane_vec_eq_r4_def: fixes v1 :: "real \<times> real \<times> real \<times> real"   and v2 :: "real \<times> real \<times> real \<times> real" shows "plane_vec_eq_r4 v1 v2 = (\<exists>k. k \<noteq> 0 \<and> v2 = k *\<^sub>s\<^sub>v\<^sub>4 v1)"
rep_eq: fixes x :: "('a, 'b) tllist" shows "tlength x = llength (fst (llist_of_tllist x, terminal x))"
homeomorphic_def: fixes s :: "'d set"   and t :: "'e set" shows "s homeomorphic t \<equiv> \<exists>f g. homeomorphism s t f g"
ValD_copy_rec_strict: fixes r :: "ValD \<rightarrow> ValD" shows "ValD_copy_rec\<cdot>r\<cdot>\<bottom> = \<bottom>"
Min_less_iff: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set"   and x :: "'k" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less (linorder.Min less_eq A) x = (\<exists>a\<in>A. less a x)"
rep_eq: shows "fpxs_nth 0 = (\<lambda>r. 0::'b)"
simps_2: fixes l :: "interval_tree"   and it :: "nat idx_interval"   and m :: "nat"   and r :: "interval_tree" shows "tree_set (Node l it m r) = {it} \<union> tree_set l \<union> tree_set r"
d: shows "M = Tensor.mat_of_cols_list (dim_row B * dim_row C) (mult.Tensor (*) (mat_to_cols_list B) (mat_to_cols_list C))"
3: shows "t \<preceq> punit.lt (rep_list p)"
aha: shows "index (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) (cs ! i) \<noteq> length (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) \<and> {xa. xa < (as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i) in fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) \<and> count_list (take (index (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) ((as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i))) (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))) xa \<le> 1} \<noteq> {}"
r_def: shows "r = run\<^sub>t (fst (\<P> \<phi>)) (fst (snd (\<P> \<phi>))) w"
simps_5: fixes b :: "bool" shows "isLam (Bool b) = False"
predI: fixes P :: "bool"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "P" shows "dynamic_component.eval tCMP active cid t t' n (dynamic_component.pred P)"
modify_tprog_fetch_action: fixes st :: "nat"   and tp :: "(action \<times> nat) list"   and b :: "nat" assumes "st \<le> length tp div 2"   and "0 < st"   and "b = 1 \<or> b = 0" shows "modify_tprog tp ! (4 * (st - Suc 0) + 2 * b) = action_map (fst (tp ! (2 * (st - Suc 0) + b)))"
ran_distinct: fixes al :: "('c \<times> 'd) list" assumes "distinct (map fst al)" shows "ran (map_of al) = snd ` set al"
assms_1: shows "m1 f|` S = m2 f|` S"
primes_omega_less_primes_omega_primorial: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "n < primorial x" shows "primes_omega n < primes_omega (primorial x)"
finsum_cong': fixes A :: "'e set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'c" assumes "A = B"   and "g \<in> B \<rightarrow> carrier M"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum M f A = finsum M g B"
defect_def: fixes N :: "'b quasinorm" shows "defect N = fst (Rep_quasinorm N)"
conv: shows "weak_conv_m (\<lambda>n. distr P borel (\<lambda>x. (g x + A.birkhoff_sum f n x) / B n)) Z"
pmf_Pi_outside: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" assumes "finite A"   and "\<exists>x. x \<notin> A \<and> f x \<noteq> dflt" shows "pmf (Pi_pmf A dflt p) f = 0"
is_IntVal_def: fixes val :: "val" shows "is_IntVal val = (\<exists>x3. val = IntVal x3)"
right_adjoint_preserved_by_iso: fixes g :: "('a S.E.term, 'a) S.arr"   and \<phi> :: "('a S.E.term, 'a) S.arr"   and g' :: "('a S.E.term, 'a) S.arr" assumes "S.is_right_adjoint g"   and "\<guillemotleft>\<phi> : g \<Rightarrow>\<^sub>S g'\<guillemotright>"   and "S.iso \<phi>" shows "S.is_right_adjoint g'"
poly_root_induct: fixes Q :: "'a poly \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and p :: "'a poly" assumes "Q 0"   and "\<And>p. (\<And>a. P a \<Longrightarrow> poly p a \<noteq> (0::'a)) \<Longrightarrow> Q p"   and "\<And>a p. \<lbrakk>P a; Q p\<rbrakk> \<Longrightarrow> Q ([:a, - (1::'a):] * p)" shows "Q p"