hcju/isabelleps huggingface dataset:
elims_2: fixes x :: "'b val" assumes "is_Intg x" obtains i :: "32 word" where "x = Intg i"
sublens_compat: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f" assumes "vwb_lens X"   and "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "X ##\<^sub>L Y"
id: shows "Polynomial.smult c (F (k - 1)) = F k * Q"
star_inductr: fixes z :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" assumes "z \<union> y O x \<subseteq> y" shows "z O x\<^sup>* \<subseteq> y"
conga_diff45: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C CongA A' B' C'" shows "A' \<noteq> B'"
lookup_mult_scalar_lt_lt: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "lookup (p \<odot> q) (punit.lt p \<oplus> lt q) = punit.lc p * lc q"
totalize_act_def: fixes act :: "('c \<times> 'c) set" shows "totalize_act act \<equiv> act \<union> Id_on (- Domain act)"
cases: fixes a1 :: "'tp"   and a2 :: "('c, 'd) lit list" assumes "a1 \<turnstile> a2" obtains \<sigma> :: "'tp" and c :: "('c, 'd) lit list" where "a1 = \<sigma>'__" and "a2 = ca__" and "infTp \<sigma>'__"   | c :: "('c, 'd) lit list" and \<sigma> :: "'tp" where "a1 = \<sigma>" and "a2 = c" and "\<forall>x\<in>nvC c. tpOfV x \<noteq> \<sigma>"
DirProds_one: fixes x :: "'e \<Rightarrow> 'f"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set" assumes "x \<in> carrier (DirProds G I)" shows "(\<forall>i\<in>I. x i = \<one>\<^bsub>G i\<^esub>) = (x = \<one>\<^bsub>DirProds G I\<^esub>)"
measure_eqI_PiM_finite: fixes I :: "'d set"   and P :: "('d \<Rightarrow> 'e) measure"   and M :: "'d \<Rightarrow> 'e measure"   and Q :: "('d \<Rightarrow> 'e) measure"   and A :: "nat \<Rightarrow> ('d \<Rightarrow> 'e) set" assumes "finite I"   and "sets P = sets (Pi\<^sub>M I M)"   and "sets Q = sets (Pi\<^sub>M I M)"   and "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure P (Pi\<^sub>E I A) = emeasure Q (Pi\<^sub>E I A)"   and "range A \<subseteq> prod_algebra I M"   and "\<Union> (range A) = space (Pi\<^sub>M I M)"   and "\<And>i. emeasure P (A i) \<noteq> \<infinity>" shows "P = Q"
ordLess_irreflexive: fixes r :: "('c \<times> 'c) set" shows "(r, r) \<notin> ordLess"
cases: fixes a :: "'b"   and r :: "('b \<times> 'c) set" assumes "a \<in> Domain r" obtains aa :: "'b" and b :: "'c" where "a = aa" and "(aa, b) \<in> r"
induced_group_homomorphism: shows "group_homomorphism induced contained.Partition ([\<cdot>]) (contained.Class \<one>) G' (\<cdot>') \<one>'"
ext_p_cost: fixes n :: "nat" shows "awalk_cost f (ext_p n) = awalk_cost f q + awalk_cost f r + real n * awalk_cost f c"
image_group_is_group: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set" assumes "weak_group_morphism f H G" shows "Group.group (image_group f G)"
lr_rep_def: shows "lr_rep \<equiv> \<lambda>r. (lr_eta_rep r, lr_theta_rep r)"
no_change_eval: fixes At :: "atomUni"   and a :: "real"   and b :: "real"   and c :: "real"   and x :: "real"   and y :: "real" assumes "getPoly At = (a, b, c)"   and "x < y"   and "\<not> (\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0)" shows "aEvalUni At x = aEvalUni At y \<and> x < y"
cone_ordLeq_cexp: fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set" assumes "cone \<le>o r1" shows "cone \<le>o r1 ^c r2"
cases: fixes x :: "('a \<times> 'a) list" obtains "x = []"   | e1 :: "'a" and e2 :: "'a" and es :: "('a \<times> 'a) list" where "x = (e1, e2) # es"
typ_of1_split_App_obtains: fixes Ts :: "typ list"   and t :: "Core.term"   and u :: "Core.term"   and ty :: "typ" assumes "typ_of1 Ts (t $ u) = Some ty" obtains uty :: "typ" where "typ_of1 Ts t = Some (uty \<rightarrow> ty)" and "typ_of1 Ts u = Some uty"
cast_TC: fixes t :: "udom defl" shows "cast\<cdot>(TC('a)\<cdot>t) = EMB(udom\<cdot>'a) oo fmapU\<cdot>(cast\<cdot>t) oo PRJ(udom\<cdot>'a)"
intensional_simps_8: fixes y :: "'b \<Rightarrow> bool" shows "#True = y \<equiv> y"
map\<^sub>0_simps_3: fixes a :: "'b" assumes "B.obj a" shows "trg\<^sub>C (\<tau>.F.map\<^sub>0 a) = \<tau>.F.map\<^sub>0 a"
CP_def: shows "CP \<equiv> {U. \<exists>\<psi> s. learn_cp \<psi> U s}"
that: fixes xs :: "'a list" assumes "f = list_succ xs"   and "set xs = orbit f s"   and "distinct xs"   and "hd xs = s" shows "thesis"
seq2: fixes c1 :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and c1' :: "('exp, 'id) MWLsCom"   and m' :: "'id \<Rightarrow> 'val"   and c2 :: "('exp, 'id) MWLsCom" assumes "SRabbr c1 m \<alpha> (Some c1') m'" shows "SRabbr (c1;c2) m \<alpha> (Some (c1';c2)) m'"
decr_maxindex: fixes p :: "poly" assumes "polybound0 p" shows "maxindex (decrpoly p) = maxindex p - 1"
real_max_mono: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c \<le> a"   and "d \<le> b" shows "max c d \<le> max a b"
commit_correct: fixes \<eta> :: "nat" shows "Schnorr_\<Sigma>_commit.abstract_com.correct \<eta>"
is_supI: fixes x :: "'a"   and s :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> s"   and "y \<sqsubseteq> s"   and "\<And>z. \<lbrakk>x \<sqsubseteq> z; y \<sqsubseteq> z\<rbrakk> \<Longrightarrow> s \<sqsubseteq> z" shows "is_sup x y s"
fine_def: fixes d :: "'c \<Rightarrow> 'd set"   and s :: "('c \<times> 'd set) set" shows "(d fine s) = (\<forall>(x, k)\<in>s. k \<subseteq> d x)"
simps_3: shows "eval_ternary_Or TernaryFalse TernaryTrue = TernaryTrue"
h5: shows "invertible_mat (M_mat (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))))"
simps_2: fixes x :: "'l"   and xs :: "'l list"   and ys :: "'l list" shows "(x # xs) @ ys = x # xs @ ys"
a_HashI: fixes v :: "Syntax.term"   and vP :: "Syntax.term"   and \<tau> :: "ty"   and h :: "hash"   and \<Gamma> :: "(Syntax.var, ty) fmap" assumes "{$$} \<turnstile> v, vP, \<lparr>vP\<rparr> : \<tau>"   and "hash \<lparr>vP\<rparr> = h"   and "value v"   and "value vP" shows "\<Gamma> \<turnstile> v, Hashed h vP, Hash h : AuthT \<tau>"
finite_enumerate_regs: fixes a :: "vname aexp" shows "finite (enumerate_regs a)"
random_oracle_def: fixes \<sigma> :: "'b \<Rightarrow> bool list option"   and x :: "'b" shows "prf.random_oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV len) \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"
h1: shows "all_list_constr_R [([], []), ([0], []), ([], [0])] (length qs)"
v3: shows "atom v3 \<sharp> (t, t', X1, v2, X4)"
thd_def: fixes tllist :: "('a, 'b) tllist" shows "thd tllist = (case tllist of TCons x21 x22 \<Rightarrow> x21)"
sv_add_invar: fixes R :: "('b \<times> 'c) set"   and I :: "'b \<Rightarrow> bool" assumes "single_valued R" shows "single_valued {(c, a). (c, a) \<in> R \<and> I c}"
bij_lens_via_comp_id_left: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'e \<Longrightarrow> 'd" assumes "wb_lens X"   and "wb_lens Y"   and "X ;\<^sub>L Y = 1\<^sub>L" shows "bij_lens X"
abcd_dcba_only_9: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c d]]" shows "\<not> [[b c d a]]"
set_disconnected_nodes_writes: fixes document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_disconnected_nodes_locs document_ptr) (set_disconnected_nodes document_ptr disc_nodes) h h'"
matrix_vector_mult_diff_distrib: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec"   and y :: "('b, 'd) vec" shows "A *v (x - y) = A *v x - A *v y"
cbig_def: fixes P :: "'prog"   and \<sigma> :: "'state" shows "cbig P \<sigma> \<equiv> {(\<sigma>'', coll). \<exists>n. (\<sigma>'', coll) \<in> csmall_nstep P \<sigma> n \<and> \<sigma>'' \<in> endset}"
C_def: shows "C = (\<lambda>x. min (B x) 1)"
comp_null_2: fixes f :: "'a" shows "f \<cdot>\<^sub>B B.null = B.null"
PfP_implies_PfP_ssubst: fixes \<beta> :: "fm"   and V :: "name set" assumes "{} \<turnstile> PfP \<guillemotleft>\<beta>\<guillemotright>"   and "V \<subseteq> Vs"   and "supp \<beta> \<subseteq> atom ` Vs" shows "{ConstP (F i) |i. i \<in> V} \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
conjugate_residuated: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "conjugation_pair f g" shows "residuated_pair f (\<lambda>x. - g (- x))"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>n l h. P (MKT n l ET h)"   and "\<And>n l v va vb vc h. P (MKT v va vb vc) \<Longrightarrow> P (MKT n l (MKT v va vb vc) h)"   and "P ET" shows "P a0"
leadsETo_induct: fixes F :: "'d program"   and za :: "'d set"   and CC :: "'d set set"   and zb :: "'d set"   and P :: "'d set \<Rightarrow> 'd set \<Rightarrow> bool" assumes "F \<in> za leadsTo[CC] zb"   and "\<And>A B. \<lbrakk>F \<in> A ensures B; A - B \<in> insert {} CC\<rbrakk> \<Longrightarrow> P A B"   and "\<And>A B C. \<lbrakk>F \<in> A leadsTo[CC] B; P A B; F \<in> B leadsTo[CC] C; P B C\<rbrakk> \<Longrightarrow> P A C"   and "\<And>B S. \<forall>A\<in>S. F \<in> A leadsTo[CC] B \<and> P A B \<Longrightarrow> P (\<Union> S) B" shows "P za zb"
assms_2: fixes n :: "nat" assumes "0 < n" shows "P n"
n_S1: fixes s :: "'s" assumes "s \<in> S1" shows "n s = (\<Sqinter>D\<in>K s. integral\<^sup>N (measure_pmf D) n)"
gt_arg_imp: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "arg t >\<^sub>t s" shows "t >\<^sub>t s"
poly_reciprocal: fixes P :: "'a poly"   and p :: "nat"   and x :: "'a" assumes "degree P \<le> p"   and "x \<noteq> (0::'a)" shows "poly (reciprocal_poly p P) x = x ^ p * poly P (inverse x)"
find_adds_SomeD1: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and u :: "'t"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "find_adds fs u = Some f" shows "f \<in> set fs"
residue_mult_group_coset: fixes b :: "nat set"   and n :: "nat"   and m :: "nat"   and m1 :: "nat"   and m2 :: "nat" assumes "b \<in> rcosets\<^bsub>residue_mult_group n\<^esub> kernel (residue_mult_group n) (residue_mult_group m) (\<lambda>k. k mod m)"   and "m1 \<in> b"   and "m2 \<in> b"   and "1 < n"   and "m dvd n" shows "m1 mod m = m2 mod m"
S_transform_Tree\<^sub>\<alpha>_Pred\<^sub>\<alpha>: fixes \<phi> :: "'c" shows "S_transform_Tree\<^sub>\<alpha> (Pred\<^sub>\<alpha> \<phi>) = Act\<^sub>\<alpha> (S_action.Pred \<phi>) (Conj\<^sub>\<alpha> bempty)"
no_volatile_Read\<^sub>s\<^sub>b_volatile_reads_consistent: fixes sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat" assumes "outstanding_refs is_volatile_Read\<^sub>s\<^sub>b sb = {}" shows "volatile_reads_consistent m sb"
prefixToLevelAppend_aux1: fixes i :: "nat"   and l :: "nat"   and a :: "('b \<times> bool) list"   and b :: "('b \<times> bool) list" assumes "i \<le> l"   and "l - i < currentLevel a" shows "prefixToLevel_aux (a @ b) l i = prefixToLevel_aux a l i"
Union_partGen: fixes P :: "'a set set" shows "\<Union> (partGen P) = \<Union> P"
nxt_inv_into: fixes q :: "hf"   and x :: "'a" assumes "q \<in> states N" shows "nxt N q x = h (nxt M (inv_into (states M) h q) x)"
td_ext_def: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and norm :: "'b \<Rightarrow> 'b" shows "td_ext Rep Abs A norm \<equiv> type_definition Rep Abs A \<and> td_ext_axioms Rep Abs norm"
smallo_imp_eventually_sgn: fixes g :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "g \<in> o(f)" shows "\<forall>\<^sub>F x in at_top. sgn (f x + g x) = sgn (f x)"
ToTC_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "T.ToTC f \<equiv> if CCC.arr f then local.Prod (local.Prod (fst f, fst (snd f)), snd (snd f)) else null"
eq_set_tree_empty: fixes t :: "('b \<times> 'c) tree" shows "(Tree2.set_tree t = {}) = (t = \<langle>\<rangle>)"
pd_ret_is_ret: fixes x :: "'n" assumes "x pd\<rightarrow> local.return" shows "x = local.return"
sturm_R: fixes p :: "real poly" shows "int (card {x. poly p x = 0}) = changes_R_smods p (pderiv p)"
Rep_FingerTree_inverse: fixes x :: "('a, 'b) FingerTree" shows "Abs_FingerTree (Rep_FingerTree x) = x"
b'_length: shows "b' < length (sorted_list_of_set J)"
OPT2_length: fixes \<sigma> :: "'a list"   and x :: "'a"   and y :: "'a" shows "length (OPT2 \<sigma> [x, y]) = length \<sigma>"
is_expansion_imp_smallo: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a"   and b :: "real \<Rightarrow> real"   and e :: "real" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "is_expansion F basis"   and "filterlim b at_top at_top"   and "\<forall>g\<in>set basis. (\<lambda>x. ln (g x)) \<in> o(\<lambda>x. ln (b x))"   and "0 < e" shows "eval F \<in> o(\<lambda>x. b x powr e)"
Oops1: fixes evsO1 :: "event list"   and A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg" assumes "evsO1 \<in> kerbIV"   and "A \<noteq> Spy"   and "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evsO1"   and "expiredAK Ta evsO1" shows "Says A Spy \<lbrace>Agent A, Agent Tgs, Number Ta, Key authK\<rbrace> # evsO1 \<in> kerbIV"
conversion_trans: fixes A :: "('b \<times> 'b) set" shows "trans (A\<^sup>\<leftrightarrow>\<^sup>*)"
semilattice_neutr_axioms: shows "semilattice_neutr (+) 0"
is_GB_cofactor_bound_def: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set"   and b :: "nat" shows "is_GB_cofactor_bound F b = (\<exists>G. punit.is_Groebner_basis G \<and> ideal G = ideal F \<and> \<Union> (indets ` G) \<subseteq> \<Union> (indets ` F) \<and> (\<forall>g\<in>G. \<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f\<in>F'. poly_deg (q f * f) \<le> b)))"
strong_fusion_intro: fixes F :: "i \<Rightarrow> bool"   and x :: "i" assumes "GEM1"   and "(\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. F z \<and> O y z))" shows "(\<sigma>x. F x) = x"
imethds_rec: fixes G :: "prog"   and I :: "qtname"   and i :: "iface" assumes "iface G I = Some i"   and "ws_prog G" shows "DeclConcepts.imethds G I = Un_tables (DeclConcepts.imethds G ` set (isuperIfs i)) \<oplus>\<oplus> (set_option \<circ> table_of (map (\<lambda>(s, mh). (s, I, mh)) (imethods i)))"
assms_1: shows "S \<equiv> trace_core_closure core1 core2 E IA IU p q"
map_add_find_right: fixes n :: "'e \<Rightarrow> 'd option"   and k :: "'e"   and xx :: "'d"   and m :: "'e \<Rightarrow> 'd option" assumes "n k = Some xx" shows "Map.map_add m n k = Some xx"
Const\<^sub>0_def: fixes c :: "'b" shows "Const\<^sub>0 c \<equiv> Poly_Mapping.single 0 c"
in_zmset_notin_mset_neg: fixes x :: "'a"   and M :: "'a zmultiset" assumes "x \<in>#\<^sub>z M"   and "x \<notin># mset_neg M" shows "x \<in># mset_pos M"
Abs_poly_mapping_inverse: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> {f. finite {x. f x \<noteq> (0::'c)}}" shows "lookup (Abs_poly_mapping y) = y"
induct: fixes P :: "'b formula \<Rightarrow> bool"   and a0 :: "'b formula" assumes "\<And>F G. P G \<Longrightarrow> P (F \<^bold>\<or> G)"   and "\<And>v. P (Atom v)"   and "P \<bottom>"   and "\<And>v. P (\<^bold>\<not> v)"   and "\<And>v va. P (v \<^bold>\<and> va)"   and "\<And>v va. P (v \<^bold>\<rightarrow> va)" shows "P a0"
lam_simps_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : hfsetcat.prod a b \<rightarrow> c\<guillemotright>" shows "local.dom (hfsetcat.\<Lambda> a b c f) = a"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>hsx x hs. P hs \<Longrightarrow> P \<langle>hsx, x, hs\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
isFDERIV_map_Var: fixes ds :: "nat list"   and N :: "nat"   and xs :: "nat list"   and vs :: "real list" assumes "length ds = N"   and "length xs = N" shows "isFDERIV N xs (map floatarith.Var ds) vs"
rename_clause_def: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'a set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.rename_clause A C = {L. \<exists>LL. LL \<in> C \<and> L = propositional_atoms.rename_literal A LL}"
morphism_sheaves_of_rings_def: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" shows "morphism_sheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"
sim_cfunc_usr: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and iu :: "'iusr_core"   and s1 :: "'s_core"   and s2 :: "'s_core'"   and ou :: "'ousr_core"   and s1' :: "'s_core"   and s2' :: "'s_core'" assumes "S p q"   and "iu \<in> IU"   and "s1 \<in> set_spmf p"   and "s2 \<in> set_spmf q"   and "(ou, s1') \<in> set_spmf (cfunc_usr core1 s1 iu)"   and "(ou, s2') \<in> set_spmf (cfunc_usr core2 s2 iu)" shows "trace_eq_simcl S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_usr core2 s2 iu)) ou)"
list_update_nonempty: fixes xs :: "'a list"   and k :: "nat"   and x :: "'a" shows "(xs[k := x] = []) = (xs = [])"
prj_in_hom_3: shows "\<guillemotleft>fgf.Prj\<^sub>0 : Chn ((f \<star> g) \<star> f) \<rightarrow>\<^sub>C f.apex\<guillemotright>"
attach_callee_def: fixes callee :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a, 'e, 'f) gpv"   and "oracle" :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b) spmf" shows "attach_callee callee oracle = (\<lambda>(s, s') q. map_spmf rprodl (exec_gpv oracle (callee s q) s'))"
progress_time_conv: fixes j :: "nat"   and \<sigma> :: "(char list \<times> 'a list) trace"   and \<sigma>' :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula" assumes "\<forall>i<j. \<tau> \<sigma> i = \<tau> \<sigma>' i" shows "Monitor.progress \<sigma> \<phi> j = Monitor.progress \<sigma>' \<phi> j"
assms_2: fixes \<sigma> :: "'\<sigma>"   and x :: "'xc"   and S :: "'xa set" assumes "c \<sigma>"   and "invar x"   and "\<alpha> x \<in> S"   and "I S \<sigma>"   and "S \<subseteq> S0"   and "\<forall>y\<in>S - {\<alpha> x}. R (\<alpha> x) y"   and "\<forall>y\<in>S0 - S. R y (\<alpha> x)" shows "I (S - {\<alpha> x}) (f x \<sigma>)"
bij_2: shows "bij_betw g {..<length ys} {..<length zs}"
fps_times_def: shows "(*) = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i = 0..n. fps_nth f i * fps_nth g (n - i)))"
abc_neq: shows "a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c"
rel_funE: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and x :: "'d"   and y :: "'e" assumes "(A ===> B) f g"   and "A x y" obtains "B (f x) (g y)"
flip_commute: fixes a :: "'b"   and b :: "'b" shows "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
junk: shows "(\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<and> (\<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<and> (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A) \<in> \<R>\<^sub>A \<and> mds\<^sub>C' = mds\<^sub>C \<and> conc.low_mds_eq mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C \<and> (\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> P"
powr_less_mono: fixes a :: "real"   and b :: "real"   and x :: "real" assumes "a < b"   and "1 < x" shows "x powr a < x powr b"
redT_updT_Some: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'h"   and xw :: "'f \<times> 'g \<Rightarrow>f nat"   and ta :: "('h, 'f, 'i) new_thread_action" assumes "ts t = \<lfloor>xw\<rfloor>"   and "thread_ok ts ta" shows "redT_updT ts ta t = \<lfloor>xw\<rfloor>"
swapEnvDom_def: fixes xs :: "'a"   and x :: "'b"   and y :: "'b"   and rho :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "swapEnvDom xs x y rho \<equiv> \<lambda>zs z. rho zs (z @zs[x \<and> y]_xs)"
confTs_map: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and vs :: "val list"   and Ts :: "ty list" shows "(P,h \<turnstile> vs [:\<le>\<^sub>\<top>] map OK Ts) = (P,h \<turnstile> vs [:\<le>] Ts)"
Field_def: fixes r :: "('d \<times> 'd) set" shows "Field r = Domain r \<union> Range r"
head_def: fixes cfg :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate" shows "head cfg = (head_com (fst cfg), snd cfg)"
sum_less_eqE: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and y :: "'c"   and x :: "'d" assumes "sum f A \<le> y"   and "x \<in> A"   and "finite A" shows "f x \<le> y"
nnegI: fixes P :: "'b \<Rightarrow> 'c" assumes "\<And>x. (0::'c) \<le> P x" shows "nneg P"
nearest_neighbors_def: fixes n :: "nat"   and p :: "(real, 'b) vec"   and kdt :: "'b kdt" shows "nearest_neighbors n p kdt = nearest_nbors n [] p kdt"
card_of_Times_infinite: fixes A :: "'c set"   and B :: "'d set" assumes "infinite A"   and "B \<noteq> {}"   and "|B| \<le>o |A|" shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
map_cong0: fixes x :: "'a set['b]"   and f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>z. z \<in> set_bset x \<Longrightarrow> f z = g z" shows "map_bset f x = map_bset g x"
inv_translation_def: fixes r :: "nat set" shows "inv_translation r \<equiv> {0..<card r} = r \<and> {0, 1} \<subseteq> r"
mult_commute: fixes a :: "'a"   and b :: "'a" shows "a \<parallel> b = b \<parallel> a"