hcju/isabelleps huggingface dataset:
filterlim_going_toI_weak: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'b filter"   and A :: "'a set" shows "filterlim f F (f going_to F within A)"
usubstappt_func_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and f :: "char"   and \<theta> :: "trm" assumes "usubstappt \<sigma> U (trm.Func f \<theta>) \<noteq> undeft" shows "usubstappt \<sigma> U \<theta> \<noteq> undeft \<and> (SFuncs \<sigma> f = undeft \<or> (\<exists>r. SFuncs \<sigma> f = Aterm r \<and> FVT r \<inter> U = {}))"
valid_sops_stmt_mono: fixes t :: "nat"   and s :: "stmt"   and t' :: "nat" assumes "valid_sops_stmt t s"   and "t \<le> t'" shows "valid_sops_stmt t' s"
is_Guard_basic_simps_9: fixes F :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "is_Guard (Guard F g c) = True"
tStep_5: shows "tObsC (t \<leadsto> s) = tObsC (t' \<leadsto> s')"
assms_2: shows "\<forall>x\<in>set ys. \<not> y \<prec> x"
set_rows_carrier: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat m n"   and "v \<in> set (rows A)" shows "v \<in> carrier_vec n"
SATAxiomI: fixes A :: "(formula list \<times> formula list set) set"   and forms :: "formula list"   and n :: "predicate"   and vs :: "vbl list"   and Gamma :: "formula list" assumes "Axioms \<subseteq> A"   and "Weaks \<subseteq> A"   and "Perms \<subseteq> A"   and "forms = [FAtom Pos n vs, FAtom Neg n vs] @ Gamma" shows "forms \<in> deductions A"
wf_chains2: fixes S :: "node fset"   and G :: "diagram"   and \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain"   and k :: "nat" assumes "S |\<subseteq>| initials G"   and "wf_dia G"   and "\<Pi> \<in> ps_chains2 S G"   and "fcard G^V + length G^E = k + fcard S" shows "wf_ps_chain \<Pi> \<and> post \<Pi> = [ terminals G |=> Bot ]"
param_dfs_variant_def: shows "param_dfs_variant \<equiv> inv_image (finite_psupset reachable <*lex*> finite_psubset <*lex*> less_than) (\<lambda>s. (dom (discovered s), pending s, length (stack s)))"
INT_insert_distrib: fixes u :: "'c"   and A :: "'c set"   and a :: "'d"   and B :: "'c \<Rightarrow> 'd set" assumes "u \<in> A" shows "(\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter> (B ` A))"
split_list_last: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
prv_imp_eqvEL: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "bprv (eqv \<phi>1 \<phi>2)" shows "bprv (imp \<phi>1 \<phi>2)"
cases: fixes x :: "'b Literal" obtains A :: "'b" where "x = Pos Aa__"   | A :: "'b" where "x = Neg A"
kbo_altdef: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "kbo'.kbo s t = (if weight_le t s then if weight_less t s then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> x = y | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> if pr_strict' (f, length ss) (g, length ts) then (True, True) else if pr_weak' (f, length ss) (g, length ts) then lex_ext_unbounded kbo'.kbo ss ts else (False, False) else (False, False))"
Rep_eqD: fixes x :: "'b word"   and y :: "'b word" assumes "to_bl x = to_bl y" shows "x = y"
Card_is_Ord: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "Ord \<kappa>"
cases: fixes a1 :: "'a multiset"   and a2 :: "'a multiset"   and ns :: "('a \<times> 'a) set" assumes "(a1, a2) \<in> multpw ns" obtains "a1 = {#}" and "a2 = {#}"   | x :: "'a" and y :: "'a" and X :: "'a multiset" and Y :: "'a multiset" where "a1 = add_mset x X" and "a2 = add_mset y Y" and "(x, y) \<in> ns" and "(X, Y) \<in> multpw ns"
simps_4: fixes x2 :: "\<sigma>"   and x1 :: "\<omega>" shows "\<sigma>\<upsilon> x2 \<noteq> \<omega>\<upsilon> x1"
Var_def: fixes x1 :: "name" shows "Var x1 \<equiv> Abs_lam (Var_Rep x1)"
that: fixes q :: "'a" assumes "raw_has_prod f N q" shows "thesis"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and RHS :: "('d, 'e) RHS" shows "P (case RHS of Branch x xa \<Rightarrow> f1 x xa | Leaf x \<Rightarrow> f2 x) = (\<not> ((\<exists>x11 x12. RHS = Branch x11 x12 \<and> \<not> P (f1 x11 x12)) \<or> (\<exists>x2. RHS = Leaf x2 \<and> \<not> P (f2 x2))))"
mod_exp_less_eq_exp: fixes a :: "int"   and n :: "nat" shows "a mod 2 ^ n < 2 ^ n"
greater_mix_pmf_greater_step_1_aux: fixes v :: "'a pmf"   and u :: "'a pmf"   and \<alpha> :: "real"   and \<beta> :: "real" assumes "v \<succeq>[\<R>] u \<and> (u, v) \<notin> \<R>"   and "\<alpha> \<in> {0<..<1}"   and "\<beta> \<in> {0<..<1}"   and "\<alpha> < \<beta>" shows "mix_pmf \<beta> v u \<succeq>[\<R>] mix_pmf \<alpha> v u \<and> (mix_pmf \<alpha> v u, mix_pmf \<beta> v u) \<notin> \<R>"
Uvariation_empty: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real" shows "Uvariation \<nu> \<nu>' {} = (\<nu> = \<nu>')"
rolling_rule: fixes g :: "'c \<rightarrow> 'b"   and f :: "'b \<rightarrow> 'c" shows "fix\<cdot>(g oo f) = g\<cdot>(fix\<cdot>(f oo g))"
free_hull_min: fixes C :: "'b list set"   and G :: "'b list set" assumes "code C"   and "G \<subseteq> \<langle>C\<rangle>" shows "\<langle>G\<rangle>\<^sub>F \<subseteq> \<langle>C\<rangle>"
lunstream_iterates_raw: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "lunstream (iterates_raw f) x = iterates f x"
list_succ_simps_3: fixes xs :: "'b list"   and x :: "'b" assumes "length xs \<le> index xs x" shows "list_succ xs x = x"
divide: shows "(R ===> R ===> R) (arith_ops_record.divide ops) (div)"
f12: shows "- c2 - (fmask2 - c2) = - fmask2"
add_closed: fixes G :: "'a set"   and g :: "'a"   and h :: "'a" assumes "AbGroup G"   and "g \<in> G"   and "h \<in> G" shows "g + h \<in> G"
conjugate_additive: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "conjugate f g" shows "additive f"
assms_1: shows "\<forall>\<^sub>F z in at z. f z = g z"
seq_stop: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val" shows "eval\<^sub>w_simple_abv (Stop ;; c, mem) (c, mem)"
TCons: fixes x :: "'a"   and xs' :: "('a, 'c) tllist"   and y :: "'b"   and ys' :: "('b, 'd) tllist" assumes "xs = TCons x xs'"   and "ys = TCons y ys'"   and "P x y"   and "tllist_all2 P Q xs' ys'" shows "thesis"
find_closest_bf_dist_take_7: fixes ps :: "(int \<times> int) list"   and p\<^sub>0 :: "int \<times> int"   and \<delta> :: "real"   and ps\<^sub>L :: "(int \<times> int) set"   and ps\<^sub>R :: "(int \<times> int) set"   and l :: "int" assumes "\<exists>p\<^sub>1\<in>set ps. dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < \<delta>"   and "distinct (p\<^sub>0 # ps)"   and "sorted_snd (p\<^sub>0 # ps)"   and "0 < length ps"   and "0 \<le> \<delta>"   and "set (p\<^sub>0 # ps) = ps\<^sub>L \<union> ps\<^sub>R"   and "\<forall>p\<in>set (p\<^sub>0 # ps). real_of_int l - \<delta> \<le> real_of_int (fst p) \<and> real_of_int (fst p) \<le> real_of_int l + \<delta>"   and "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"   and "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"   and "sparse \<delta> ps\<^sub>L"   and "sparse \<delta> ps\<^sub>R" shows "\<forall>p\<^sub>1\<in>set ps. dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case find_closest_bf p\<^sub>0 (take 7 ps) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))"
S2EnvUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [ENext p]_(e p) \<and> $S2 rmhist p \<longrightarrow> unchanged e p"
rel_compp: fixes R :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and S :: "'g \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun (=) (R OO S) = rel_fun (=) R OO rel_fun (=) S"
ag_eq_diffzero: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e"   and b :: "'e" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A" shows "(a = b) = (a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> b = \<zero>\<^bsub>A\<^esub>)"
inv_comp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "category.iso C f"   and "category.iso C g"   and "partial_magma.arr C (C g f)" shows "category.inv C (C g f) = C (category.inv C f) (category.inv C g)"
nonzero_mult_divide_mult_cancel_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times c b) = divide a b"
B_def: shows "B = (\<lambda>i. range (A (x i)) \<union> {UNIV})"
uint_msb_test_bit: fixes x :: "uint" shows "msb x = bit x wivs_index"
convex_def: fixes s :: "'c set" shows "convex s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
simps_2: fixes uu :: "'b" shows "normalized_nnf_match (Match uu) = True"
adopt_node_locs_def: shows "Shadow_DOM.adopt_node_locs \<equiv> l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs Shadow_DOM.remove_child_locs set_disconnected_nodes_locs"
has_integral_component_nonneg: fixes k :: "'a"   and f :: "'b \<Rightarrow> 'a"   and i :: "'a"   and S :: "'b set" assumes "k \<in> Basis"   and "(f has_integral i) S"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x \<bullet> k" shows "0 \<le> i \<bullet> k"
inverse_less_iff_less: 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"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a"   and "less zero b" shows "less (inverse a) (inverse b) = less b a"
sinks_cons_same: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list" assumes "refl I" shows "sinks I D (D x) (x # xs) = insert (D x) (sinks I D (D x) xs)"
wp2_is_pre: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "\<turnstile>\<^sub>2 {wp\<^sub>2 c Q} c {Q}"
td_UBound_i: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" assumes "finite PROB" shows "{card (traversed_states (fst p) (snd p)) - 1 |p. fst p \<in> valid_states PROB \<and> snd p \<in> valid_plans PROB} \<noteq> {}"
path_image_flow_to_path: fixes t :: "real"   and t' :: "real"   and x :: "'a" assumes "t \<le> t'" shows "path_image (flow_to_path x t t') = flow0 x ` {t..t'}"
frameImpClosed: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and F :: "'d frame"   and \<Phi> :: "'e"   and p :: "(name \<times> name) list" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.FrameImp SImp F \<Phi>" shows "assertionAux.FrameImp SImp (p \<bullet> F) (p \<bullet> \<Phi>)"
invariant_1D_3: fixes plr :: "int poly \<times> rat \<times> rat" assumes "invariant_1 plr" shows "sgn (real_of_1 plr) = real_of_rat (sgn (rai_ub plr))"
many_strongly_connected_implies_12_eq: fixes x :: "'a" assumes "many_strongly_connected x" shows "x\<^sup>\<star> ; x ; x\<^sup>T = x\<^sup>+"
l_get_owner_document_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_owner_document_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs ShadowRootClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs DocumentClass.known_ptr Shadow_DOM.get_parent Shadow_DOM.get_parent_locs get_root_node_si get_root_node_si_locs CD.a_get_owner_document get_host get_host_locs Shadow_DOM.get_owner_document get_shadow_root get_shadow_root_locs get_tag_name Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs ShadowRootClass.known_ptrs get_ancestors_si get_ancestors_si_locs"
path_join_path_ends: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "path (g1 +++ g2)"   and "path g2" shows "pathfinish g1 = pathstart g2"
mult_minus_eq_nat: fixes x :: "nat"   and y :: "nat"   and z :: "nat" assumes "x + y = z" shows "- int x - int y = - int z"
simps_4: fixes C :: "('a, 'b, 'c) psiCase"   and xvec :: "name list"   and Tvec :: "'a list" shows "subs (Case C) xvec Tvec = Case subs'' C xvec Tvec"
map_simp: fixes f :: "'j" assumes "J.arr f" shows "\<chi>.A.map f = a"
assms: shows "\<forall>res. \<not> evaluate_prog False env s prog res"
ide_MkIde: fixes A :: "'a E.term" assumes "A \<in> IDE" shows "ide (MkIde A)"
refine: fixes Xi :: "'b list"   and X :: "'c set"   and Di :: "nat"   and D :: "nat" assumes "(Xi, X) \<in> appr_rel"   and "(Di, D) \<in> nat_rel"   and "DIM_precond TYPE('c) D" shows "(nres_of (mig_set_impl Di Xi), mig_set $ D $ X) \<in> \<langle>rnv_rel\<rangle>nres_rel"
K0_stable: fixes B :: "'a set" assumes "B \<in> K0" shows "induced_map A -` B \<in> K0"
Live_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "Live P = (\<lambda>s. P s \<or> \<not> M\<^sub>p P s)"
chain_fun: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "('a \<Rightarrow> 'b) set"   and a :: "'a" assumes "Complete_Partial_Order.chain (fun_ord ord) A" shows "Complete_Partial_Order.chain ord {y. \<exists>f\<in>A. y = f a}"
the_def: fixes option :: "'a option" shows "the option = (case option of Some x2 \<Rightarrow> x2)"
mono_extend_set_offending_flows: fixes m :: "'b SecurityInvariant_configured"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set"   and F' :: "('b \<times> 'b) set" assumes "configured_SecurityInvariant m"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E"   and "F' \<in> c_offending_flows m \<lparr>nodes = V, edges = E'\<rparr>" shows "\<exists>F\<in>c_offending_flows m \<lparr>nodes = V, edges = E\<rparr>. F' \<subseteq> F"
is_Zp_cauchy_imp_closed: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "padic_integers p"   and "padic_integers.is_Zp_cauchy p s" shows "s \<in> carrier padic_int p\<^bsup>\<omega>\<^esup>"
Cons_path: assumes "g \<turnstile> hd (tl ns)-tl ns\<rightarrow>m"   and "n \<in> set (predecessors g (hd (tl ns)))" shows "thesis"
l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf \<equiv> type_wf = ShadowRootClass.type_wf"
IH: fixes \<phi> :: "'a ltl"   and w' :: "nat \<Rightarrow> 'a set" assumes "\<forall>\<psi>\<in>\<^bold>G \<phi>. w' \<Turnstile> \<psi> = (xs \<frown> w') \<Turnstile> \<psi>" shows "(xs \<frown> w') \<Turnstile> \<phi> = w' \<Turnstile> af\<^sub>G \<phi> xs"
well_def_step_R: fixes subsets1 :: "(nat list \<times> nat list) list"   and qs1 :: "real poly list"   and subsets2 :: "(nat list \<times> nat list) list"   and qs2 :: "real poly list" assumes "all_list_constr_R subsets1 (length qs1)"   and "all_list_constr_R subsets2 (length qs2)" shows "all_list_constr_R (subsets_smash_R (length qs1) subsets1 subsets2) (length (qs1 @ qs2))"
assms: shows "segment G.face_cycle_succ a' a \<noteq> {} \<or> segment G.face_cycle_succ a a' \<noteq> {}"
Inverse_relSym: fixes f :: "'b"   and g :: "'b" assumes "cinv f g" shows "cinv g f"
Cong_perm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D" shows "Cong A B C D \<and> Cong A B D C \<and> Cong B A C D \<and> Cong B A D C \<and> Cong C D A B \<and> Cong C D B A \<and> Cong D C A B \<and> Cong D C B A"
StepI: shows "exec_step_input P C M pc ics = StepI i"
step_3: fixes s :: "'c"   and vs :: "'a \<Rightarrow> nat"   and ws :: "'b \<Rightarrow> nat" assumes "y = (s, vs)"   and "vs = ws \<circ> g" shows "((f s, ws), f t, \<lambda>x. if x \<in> range g then vs' (inv g x) else ws x) \<in> (step (map_minsky f g M))\<^sup>*"
is_stmt_def: fixes t :: "Term.term" shows "is_stmt t = (\<exists>c. t = In1r c)"
whr_inv_auto_1: fixes t :: "trm"   and t' :: "trm" assumes "t \<leadsto> t'" obtains x :: "name" and t\<^sub>1 :: "trm" and t\<^sub>2 :: "trm" where "t = App (Lam [x].t\<^sub>1__) t\<^sub>2__" and "t' = t\<^sub>1__[x::=t\<^sub>2__]"   | t\<^sub>1 :: "trm" and t\<^sub>1' :: "trm" and t\<^sub>2 :: "trm" where "t = App t\<^sub>1 t\<^sub>2" and "t' = App t\<^sub>1' t\<^sub>2" and "t\<^sub>1 \<leadsto> t\<^sub>1'"
exp_add: fixes x :: "'d"   and y :: "'d" shows "exp (x + y) = exp x * exp y"
HARootStates_Range_ChiPlus: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "S \<in> States (HARoot A)" shows "S \<notin> Range (ChiPlus A)"
msed_map_invR: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset"   and b :: "'c"   and N :: "'c multiset" assumes "image_mset f M = add_mset b N" shows "\<exists>M1 a. M = add_mset a M1 \<and> f a = b \<and> image_mset f M1 = N"
intro: fixes cstep1 :: "'a \<Rightarrow> 'b + 'c \<Rightarrow> 'a \<Rightarrow> bool"   and en1 :: "'d \<times> 'e \<Rightarrow> 'b \<Rightarrow> bool option"   and ex1 :: "'d \<times> 'e \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'e) option"   and cstep2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and en2 :: "'d \<times> 'e \<Rightarrow> 'b \<Rightarrow> bool"   and ex2 :: "'d \<times> 'e \<Rightarrow> 'b \<Rightarrow> 'd \<times> 'e"   and wf_local :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and ginit :: "('a, 'd, 'e) Gen_Scheduler.global_config set" assumes "sched_typing cstep1 en1 ex1 cstep2 en2 ex2 wf_local"   and "sched_typing_init_axioms ginit wf_local" shows "sched_typing_init cstep1 en1 ex1 cstep2 en2 ex2 ginit wf_local"
prefixBeforeElementDoesNotContainTrailElement: fixes e :: "'b"   and M :: "('b \<times> bool) list" assumes "e \<in> set (elements M)" shows "e \<notin> set (elements (prefixBeforeElement e M))"
ord_of_eq_0_iff: fixes n :: "nat" shows "(ord_of_nat n = 0) = (n = 0)"
handshake_phase_invD: fixes s :: "'mut process_name \<Rightarrow> ('d, 'mut, 'e, 'f) Model.local_state" assumes "handshake_phase_inv s" shows "(sys_ghost_hs_in_sync m s, sys_hs_type s, sys_ghost_hs_phase s, mut_ghost_hs_phase s) \<in> hp_step_rel \<and> (sys_hs_pending m s \<longrightarrow> \<not> sys_ghost_hs_in_sync m s)"
bounded_quasi_semilattice_axioms: shows "bounded_quasi_semilattice lcm (1::'a) (0::'a) normalize"
xor_self: fixes x :: "'b" shows "x XOR x = (0::'b)"
cod_mkArr: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and B :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and F :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr (mkArr A B F)" shows "cod (mkArr A B F) = mkIde B"
pos_zmult_eq_1_iff: fixes m :: "int"   and n :: "int" assumes "0 < m" shows "(m * n = 1) = (m = 1 \<and> n = 1)"
map\<^sub>0_simps_4: fixes a :: "'c" assumes "C.obj a" shows "D.dom (F.map\<^sub>0 a) = F.map\<^sub>0 a"
hyps_2: fixes v :: "int" assumes "paval' a2 ps = Some v" shows "vars a2 \<subseteq> dom ps"
mod_2_is_both_even_or_odd: fixes i :: "'a"   and j :: "'a" shows "(even i \<and> even j \<or> odd i \<and> odd j) = (i mod (2::'a) = j mod (2::'a))"
wr2: shows "w2 \<equiv>\<^sub>E r'\<^sub>2"
ACall_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('c, 'd) chan"   and p :: "PrIds"   and v :: "state \<Rightarrow> 'c" shows "Call ch p v \<equiv> \<not> $Calling ch p \<and> cbit<ch!p>$ \<noteq> $rbit<ch!p> \<and> arg<ch!p>$ = $v"
set_less_aux_code: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "ord.set_less_aux less_eq (set xs) (set ys) = ord.lexordp (\<lambda>x y. less y x) xs ys"
seqSubst4: fixes y :: "name"   and \<sigma> :: "(name list \<times> 'a list) list" assumes "y \<sharp> \<sigma>"   and "wellFormedSubst \<sigma>" shows "nameTerm y[<\<sigma>>] = nameTerm y"
nsc_of_atom: fixes v :: "nat \<Rightarrow> 'a"   and a :: "'a atom" shows "v \<Turnstile>\<^sub>n\<^sub>s nsc_of_atom a = v \<Turnstile>\<^sub>a a"
lemma_2_10_25: fixes a :: "'a"   and b :: "'a" shows "a \<le> (a r\<rightarrow> b) l\<rightarrow> a"
transfer: shows "rel_fun pcr_word (=) (\<lambda>k n. n < LENGTH('b) \<and> bit k n) bit"
conj: fixes F :: "('d \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "is_filter F"   and "F P"   and "F Q" shows "F (\<lambda>x. P x \<and> Q x)"
Infinitesimal_FreeUltrafilterNat_iff: fixes X :: "nat \<Rightarrow> 'b" shows "(star_n X \<in> Infinitesimal) = (\<forall>u>0. \<forall>\<^sub>F n in \<U>. norm (X n) < u)"
rep_Gromov_completion_to_Gromov_completion: fixes y :: "'a" shows "rep_Gromov_completion (to_Gromov_completion y) = (\<lambda>n. y)"
inject_1: fixes a :: "'b"   and a' :: "'b" shows "(Inl a = Inl a') = (a = a')"
module_on_axioms: fixes p :: "'a" shows "module_on (src.tangent_space p) (*\<^sub>R)"
get_option_def: fixes f :: "'s \<Rightarrow> ('a, 'm) optionT" shows "local.get_option f = OptionT (get (\<lambda>s. run_option (f s)))"
trss_add_context_s: fixes s :: "'f list"   and c :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and ce :: "'f list multiset" assumes "((s, c), e, s', c') \<in> trss fg"   and "mon_e fg e \<inter> mon_c fg ce = {}" shows "((s, c + ce), e, s', c' + ce) \<in> trss fg"
vec_nth_inject: fixes x :: "('b, 'a) Finite_Cartesian_Product.vec"   and y :: "('b, 'a) Finite_Cartesian_Product.vec" shows "(($h) x = ($h) y) = (x = y)"
8: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<squnion> z = x \<squnion> (y \<squnion> z)"