hcju/isabelleps huggingface dataset:
Cons_1: fixes state :: "State"   and newWl :: "nat list" assumes "getM state = M @ [(opposite literal, decision)]"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"   and "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)"   and "uniq Wl'" shows "let state' = notifyWatches_loop literal Wl' newWl state in InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
xpos: shows "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y \<in> C"
Glb_lower: fixes L :: "'b set"   and Glb :: "'b set \<Rightarrow> 'b"   and A :: "'b set"   and a :: "'b" assumes "Complete_Lattice L Glb"   and "A \<subseteq> L"   and "a \<in> A" shows "Glb A \<le> a"
norm_stack_Cons: fixes d :: "('dyn, 'ubx1, 'ubx2) unboxed"   and \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list" shows "norm_stack (d # \<Sigma>) = Subx.norm_unboxed d # norm_stack \<Sigma>"
iwlsFSwSTR_qInitAbs_respectsP_alphaAbs: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSwSTR MOD" shows "qInitAbs MOD respectsP ($=)"
free_decompose: shows "\<forall>ts v. 0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot) \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> (0 < \<parallel>ext va\<parallel> \<and> |lan va| = 1 \<and> (\<forall>c. \<parallel>len va ts c\<parallel> = 0 \<or> restrict va (clm ts) c = bot \<and> restrict va (res ts) c = bot)) \<and> 0 < \<parallel>ext u\<parallel> \<and> |lan u| = 1 \<and> (\<forall>c. \<parallel>len u ts c\<parallel> = 0 \<or> restrict u (clm ts) c = bot \<and> restrict u (res ts) c = bot))"
iMOD_0_finite: fixes r :: "nat" shows "finite [ r, mod 0 ]"
inverse: fixes h :: "'a \<Rightarrow> real" assumes "h \<in> H" shows "(\<lambda>x. inverse (h x)) \<in> H"
mat_geD: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat"   and i :: "nat"   and j :: "nat" assumes "ord.mat_ge less_eq A B"   and "i < dim_row A"   and "j < dim_col A" shows "less_eq (B $$ (i, j)) (A $$ (i, j))"
eta_preserves_typ_of: fixes t :: "Core.term"   and t' :: "Core.term"   and \<tau> :: "typ" assumes "t \<rightarrow>\<^sub>\<eta> t'"   and "typ_of t = Some \<tau>" shows "typ_of t' = Some \<tau>"
simps_1: fixes c1 :: "'a acom"   and c2 :: "'a acom" shows "sub\<^sub>1 (c1;; c2) = c1"
I: fixes X :: "'a set"   and x :: "'a"   and y :: "'a"   and a :: "'a"   and b :: "real" assumes "X \<in> \<F>'"   and "x \<in> X"   and "y \<in> X"   and "(a, b) \<in> I" shows "a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or> b \<le> a \<bullet> x \<and> b \<le> a \<bullet> y"
takeWhile_append1: fixes x :: "'b"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b list" assumes "x \<in> set xs"   and "\<not> P x" shows "takeWhile P (xs @ ys) = takeWhile P xs"
assms_6: shows "moebius_pt M1 z3 = w3"
DeltaNegExists: fixes A :: "('a, 'b) form"   and n :: "'a"   and G :: "('a, 'b) form list" assumes "\<turnstile> Neg (A[App n []/0]) # G"   and "news n (A # G)" shows "\<turnstile> Neg (Exists A) # G"
round_up_shift: fixes p :: "int"   and x :: "real"   and k :: "int" shows "round_up p (x * 2 powr real_of_int k) = 2 powr real_of_int k * round_up (p + k) x"
less: fixes y :: "nat" assumes "y < n" shows "f y \<le> C\<^sub>3 * real y + C\<^sub>4"
FunctorCompM: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and f :: "'c"   and X :: "'a"   and Y :: "'a" assumes "FunctorM F"   and "f maps\<^bsub>CatDom F\<^esub> X to Y" shows "F ## f maps\<^bsub>CatCod F\<^esub> F @@ X to F @@ Y"
geom_spick_eq: fixes m :: "nat"   and w :: "bool stream"   and x :: "bool" shows "geom_proc m (spick w m x) = geom_proc m w"
existence_ivl_trans': fixes t :: "real"   and s :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t + s - t0 \<in> existence_ivl0 x0"   and "t - t0 \<in> existence_ivl0 x0" shows "s \<in> existence_ivl0 (flow0 x0 (t - t0))"
iT_arith_iAll_conv_4: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t k \<ominus> I. P t) = (\<box> t I \<down>\<le> k. P (k - t))"
reflect_poly_mult: fixes p :: "'a poly"   and q :: "'a poly" shows "reflect_poly (p * q) = reflect_poly p * reflect_poly q"
all_security_requirements_fulfilled_ACS_D: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" assumes "all_security_requirements_fulfilled M G" shows "all_security_requirements_fulfilled (get_ACS M) G"
stone_phi_embed_homomorphism: shows "(bounded_lattice_homomorphism stone_phi_embed \<and> uminus_homomorphism stone_phi_embed) \<and> (\<forall>x y. x \<le> y \<longrightarrow> stone_phi_embed x \<le> stone_phi_embed y)"
prj_chine_assoc'_3: shows "Prj\<^sub>0 \<cdot> chine_assoc' = Prj\<^sub>0\<^sub>0"
inj_map: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f" shows "inj ((\<circ>) f)"
weakPsiCongBangPres: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "\<forall>\<Psi>. \<Psi> \<rhd> P \<doteq> Q"   and "guarded P"   and "guarded Q" shows "\<Psi> \<rhd> !P \<doteq> !Q"
entails_ex_post: fixes P :: "assn"   and Q :: "'c \<Rightarrow> assn" assumes "\<not> (P \<Longrightarrow>\<^sub>A \<exists>\<^sub>Ax. Q x)" shows "\<forall>x. \<not> (P \<Longrightarrow>\<^sub>A Q x)"
One_not_eq_fromArr: fixes Arr :: "'a set" shows "quiver.One \<notin> quiver.fromArr Arr ` Arr"
components_are_iso: fixes a :: "'a" assumes "B.ide a" shows "B.iso (B.R a)"
wls_swapEnv_involutive: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "wlsEnv rho" shows "rho &[x \<and> y]_zs &[x \<and> y]_zs = rho"
id_act_3: fixes v :: "i"   and \<alpha> :: "'b"   and \<beta> :: "'b" shows "[\<alpha> \<^bold>= \<beta> \<^bold>\<equiv> \<^bold>\<A>(\<alpha> \<^bold>= \<beta>) in v]"
match_supp_snd: fixes p :: "pat"   and u :: "trm"   and \<theta> :: "(name \<times> trm) list" assumes "\<turnstile> p \<rhd> u \<Rightarrow> \<theta>" shows "supp (map snd \<theta>) = supp u"
\<omega>consistentStd2_def: shows "\<omega>consistentStd2 \<equiv> \<forall>\<phi>\<in>fmla. \<forall>x\<in>var. Fvars \<phi> = {x} \<longrightarrow> (\<forall>n\<in>num. prv (subst \<phi> n x)) \<longrightarrow> \<not> prv (exi x (neg \<phi>))"
assms_2: shows "pmdl (set gs) = pmdl (set (init_syzygy_list bs))"
gram_schmidt0_result_4: fixes ws :: "'b vec list"   and n :: "nat" assumes "set ws \<subseteq> carrier_vec n" shows "LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set ws) = LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set (gram_schmidt0 n ws))"
det_row_add_iterate: fixes A :: "(('b, 'c) vec, 'c) vec"   and i :: "'c"   and j :: "'c" shows "det (row_add_iterate A (nrows A - 1) i j) = det A"
D: shows "\<exists>v\<in>sources_aux I D U zs. (D x, v) \<in> I"
RETURN_as_SPEC_refine: fixes M :: "'a nres"   and a :: "'b"   and R :: "('a \<times> 'b) set" assumes "M \<le> SPEC (\<lambda>c. (c, a) \<in> R)" shows "M \<le> \<Down> R (RETURN a)"
segment_Worder: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Worder D" shows "Worder (Iod D (segment D a))"
add_le_imp_le_diff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and i :: "'b"   and k :: "'b"   and n :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (plus i k) n" shows "less_eq i (minus n k)"
basic: fixes f :: "'b \<Rightarrow> 'b"   and P :: "'b set" shows "\<turnstile> \<lbrace>\<acute>f \<in> P\<rbrace> Basic f P"
project_constrains_I: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set"   and C :: "'c set" assumes "extend h F \<squnion> G \<in> extend_set h A co extend_set h B" shows "F \<squnion> project h C G \<in> A co B"
cases: fixes x :: "tm \<times> tm \<times> tm" obtains l :: "name" and s :: "tm" and k :: "tm" and sl :: "name" and y :: "tm" where "atom l \<sharp> (s, k, sl)" and "atom sl \<sharp> s" and "x = (s, k, y)"
send_correct: fixes i :: "'a \<times> nat"   and m :: "('a \<times> nat, 'b) message" assumes "event_at i (Send m)" shows "(\<exists>n \<sigma>. Inr m = state i \<bind> (\<lambda>s. create_insert s n \<sigma> i)) \<or> (\<exists>n. Inr m = state i \<bind> (\<lambda>s. create_delete s n))"
complex_derivative_mult_at: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and g :: "complex \<Rightarrow> complex" assumes "f analytic_on {z}"   and "g analytic_on {z}" shows "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
div_unit_dvd_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times (divide a b) c = dvd.dvd times a c"
monomials_and_monoms: fixes k :: "nat \<Rightarrow>\<^sub>0 nat"   and m :: "'a mpoly" shows "(k \<in> monomials m) = (\<exists>a. a \<noteq> 0 \<and> monomials (MPoly_Type.monom k a) \<subseteq> monomials m)"
prems_6: shows "nprv (insert \<psi> G \<union> F) \<chi>"
finiteUn: fixes A :: "'b set"   and B :: "'b set" shows "finite (A \<union> B) = (finite A \<and> finite B)"
arctan_add_small: fixes x :: "real"   and y :: "real" assumes "\<bar>x * y\<bar> < 1" shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
adm_wo_def: fixes H :: "('a Bonk_Schramm_extension_unfolded \<Rightarrow> 'b) \<Rightarrow> 'a Bonk_Schramm_extension_unfolded \<Rightarrow> 'b" shows "adm_wo Bonk_Schramm_extension_unfolded_wo H \<equiv> \<forall>f g x. (\<forall>y\<in>underS Bonk_Schramm_extension_unfolded_wo x. f y = g y) \<longrightarrow> H f x = H g x"
Compl_atLeast: fixes k :: "'c" shows "- {k..} = {..<k}"
identity_2: fixes g :: "'a" assumes "g \<in> G" shows "binop g e = g"
assms_2: fixes x :: "'b" assumes "x \<in> outs_gpv (\<I>_uniform UNIV {x. Domainp R x}) gpv" shows "C x x"
match_subst_closed: fixes pat :: "term"   and t :: "'a"   and env :: "(name, 'a) fmap"   and rhs :: "'a" assumes "match pat t = Some env"   and "closed_except rhs (frees pat)"   and "closed t" shows "closed (subst rhs env)"
convcomp_inv: fixes c :: "'d \<Rightarrow>\<^sub>U 'c" shows "c \<circ>\<^sub>C inv\<^sub>C c = id\<^sub>C"
fields_def2: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs" shows "fields P C = FDTs"
le_add2: fixes n :: "nat"   and m :: "nat" shows "n \<le> m + n"
E20: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>F \<longrightarrow> \<box>\<diamond>F"
simps_2: fixes x :: "udom"   and f :: "udom \<rightarrow> udom\<cdot>'a error" shows "bindU\<cdot>(Ok\<cdot>x)\<cdot>f = f\<cdot>x"
semiring_normalization_rules_23: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "plus (plus a b) c = plus (plus a c) b"
sat_Since_rec: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "formula"   and I :: "\<I>"   and \<psi> :: "formula" shows "sat \<sigma> V v i (Since \<phi> I \<psi>) = (mem 0 I \<and> sat \<sigma> V v i \<psi> \<or> 0 < i \<and> enat (\<Delta> \<sigma> i) \<le> right I \<and> sat \<sigma> V v i \<phi> \<and> sat \<sigma> V v (i - 1) (Since \<phi> (subtract (\<Delta> \<sigma> i) I) \<psi>))"
distinct_3: fixes x3 :: "bool" shows "Unit \<noteq> Bool x3"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "set (map f xs) = f ` set xs"
neg_cor_neg_one: shows "- cor (- 1) = 1"
smult_l_minus: fixes a :: "'a"   and x :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "x \<in> carrier P" shows "\<ominus> a \<odot>\<^bsub>P\<^esub> x = \<ominus>\<^bsub>P\<^esub> (a \<odot>\<^bsub>P\<^esub> x)"
assms_2: fixes i :: "nat" shows "Sigma_Algebra.measure M (c i) = 0"
mat_O_dim: shows "mat_O \<in> carrier_mat N N"
res_fold_A: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "ChamberComplexFolding A (restrict1 (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) (\<Union> A))"
castVar_asSort_Var: fixes xs :: "'varSort"   and x :: "'var" shows "castVar (asSort xs) (Var xs x) = (xs, x)"
length_list_of: fixes xs :: "'b llist" assumes "lfinite xs" shows "enat (length (list_of xs)) = llength xs"
ord_less_eq_trans: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" assumes "a <s b"   and "b = c" shows "a <s c"
sel_1: fixes s1 :: "'b stream"   and s2 :: "'c stream" shows "shd (s1 ||| s2) = (shd s1, shd s2)"
negligible_lowdim: fixes S :: "'c set" assumes "dim S < DIM('c)" shows "negligible S"
conj_xor_distrib: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x AND (y XOR z) = x AND y XOR x AND z"
il_less_eq: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<squnion> y = y)"
simps_3: fixes B :: "'a set"   and C :: "'a set"   and Cs :: "'a set list" shows "wall_crossings (B # C # Cs) = THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}}) \<and> separated_by H B C) # wall_crossings (C # Cs)"
trans: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and R :: "('procS, 'procT) Proc" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> Q"   and "Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> R" shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> R"
blinding_hash_eq: fixes x :: "'a"   and y :: "'a" assumes "bo x y" shows "h x = h y"
prv_exi_PPf_iff_isTrue: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and enc :: "'f \<Rightarrow> 'e"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and P :: "'f"   and isTrue :: "'f \<Rightarrow> bool"   and Pf :: "'f"   and \<phi> :: "'f" assumes "Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (exi (Generic_Syntax.Variable var 11) (Minimal_Truth_Soundness_HBL1iff_Compl_Pf.PPf var Var FvarsT Fvars subst Pf (Var (Generic_Syntax.Variable var 11)) (enc \<phi>))) = isTrue (exi (Generic_Syntax.Variable var 11) (Minimal_Truth_Soundness_HBL1iff_Compl_Pf.PPf var Var FvarsT Fvars subst Pf (Var (Generic_Syntax.Variable var 11)) (enc \<phi>)))"
sets: fixes a :: "'b" assumes "a \<in> space M" shows "sets (K a) = sets N"
simps_3: fixes b :: "nat"   and f :: "nat \<Rightarrow> Ord" shows "evalO b (L f) = evalO b (f b)"
reduced_GB_D1: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "punit.is_reduced_GB G" shows "punit.is_Groebner_basis G"
invar_def: fixes g :: "'m" shows "invar g \<equiv> M_invar g \<and> (\<forall>l\<in>ran (M_lookup g). distinct (map snd l)) \<and> finite (WGraph.edges (\<alpha> g))"
mem_Joints_msum_aforms'E: fixes xa :: "real list"   and x :: "(real \<times> real pdevs) list"   and y :: "(real \<times> real pdevs) list" assumes "xa \<in> Joints (msum_aforms' x y)" obtains a :: "real list" and b :: "real list" where "xa = map2 (+) a b" and "a \<in> Joints x" and "b \<in> Joints y"
Cons: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "x < y" shows "lexordp_eq (x # xs) (y # ys)"
special_solutions_def: fixes a :: "nat list"   and b :: "nat list" shows "special_solutions a b = concat (map (\<lambda>i. map (sij a b i) [0..<length b]) [0..<length a])"
inj_on_imp_Ker0: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T"   and "inj_on T V" shows "ker T \<inter> V = 0"
intro: fixes G :: "'b \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "G \<circ> F = identity_functor.map B"   and "F \<circ> G = identity_functor.map A" shows "inverse_functors_axioms A B F G"
\<R>_typed_step_plus: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<leadsto>\<^sup>+ \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>"   and "\<turnstile> \<Gamma>,\<S>,P {c\<^sub>1} \<Gamma>',\<S>',P'"   and "no_await c\<^sub>1"   and "tyenv_wellformed mds \<Gamma> \<S> P"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"   and "pred P mem\<^sub>1"   and "pred P mem\<^sub>2"   and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x" shows "\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>1, mds, mem\<^sub>2\<rangle> \<leadsto>\<^sup>+ \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>"
app_const_distinct: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and u\<^sub>1 :: "'c"   and u\<^sub>2 :: "'c"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "app u\<^sub>1 u\<^sub>2 \<noteq> const name"
PiE_mem: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'d set"   and T :: "'d \<Rightarrow> 'e set"   and x :: "'d" assumes "f \<in> Pi\<^sub>E S T"   and "x \<in> S" shows "f x \<in> T x"
gcd_poly_code_aux_correct: fixes p :: "'a poly"   and q :: "'a poly" assumes "content p = (1::'a)"   and "q = 0 \<or> content q = (1::'a)" shows "gcd_poly_code_aux p q = gcd p q"
list_tail_coinc: fixes n1 :: "'e"   and r1 :: "'e list"   and n2 :: "'e"   and r2 :: "'e list" assumes "n1 # r1 = n2 # r2" shows "n1 = n2 \<and> r1 = r2"
assms_1: fixes dir :: "('i, 'a) Direction"   and x :: "nat"   and c :: "'a" assumes "dir = Positive \<or> dir = Negative"   and "a = LE dir x c"   and "\<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)" shows "P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) s"
dualization_rew_1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<not> \<box>F \<equiv> \<diamond>\<not> F"
other: assumes "unfree t = None"   and "unapp t = None"   and "unconst t = None" shows "thesis"
measurable_CI: shows "M.I \<in> borel_measurable (restrict_space borel {0<..<1})"
to_function_def: fixes S :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" shows "to_function S f = (\<lambda>s\<in>carrier S. eval S S (\<lambda>x. x) s f)"
not_Ici_eq_empty: fixes l :: "'a" shows "{l..} \<noteq> {}"
smap_szip_snd: fixes g :: "'e \<Rightarrow> 'c"   and s1 :: "'d stream"   and s2 :: "'e stream" shows "smap (\<lambda>x. g (snd x)) (s1 ||| s2) = smap g s2"
that: fixes a' :: "'edge" assumes "valid_edge a'"   and "sourcenode a' = sourcenode a"   and "targetnode a' \<notin> N" shows "thesis"
prod_diff1: fixes A :: "'a set"   and f1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b" assumes "finite A" shows "(\<Prod>x\<in>A. f1 x - f2 x) = (\<Sum>X\<in>Pow A. (- (1::'b)) ^ card X * prod f2 X * prod f1 (A - X))"
fold_Option_bind_eq_Some_at_point_not_None': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a option"   and l1 :: "('b \<times> 'c) list"   and l2 :: "('b \<times> 'c) list"   and start :: "'a option"   and res :: "'a" assumes "fold (\<lambda>(x, y) option. Option.bind option (f x y)) (l1 @ l2) start = Some res" shows "fold (\<lambda>(x, y) option. Option.bind option (f x y)) l1 start \<noteq> None"
x_neq_xs: shows "\<forall>v\<in>set xs. x \<noteq> v"
smult_add_left: fixes a :: "'a"   and b :: "'a"   and p :: "'a poly" shows "Polynomial.smult (a + b) p = Polynomial.smult a p + Polynomial.smult b p"
recvmsg_def: fixes P :: "'e \<Rightarrow> bool"   and a :: "'e seq_action" shows "recvmsg P a \<equiv> case a of receive m \<Rightarrow> P m | _ \<Rightarrow> True"
Ns_def: shows "Ns = {normal_vector W |W. cond W \<and> (\<forall>w\<in>X. (0::'a) \<le> normal_vector W \<bullet> w)} \<union> {- normal_vector W |W. cond W \<and> (\<forall>w\<in>X. (0::'a) \<le> - normal_vector W \<bullet> w)}"
simps: fixes I :: "('b, 'c, 'a) interp"   and i :: "'b" shows "FunctionFrechet I i = (THE f'. \<forall>x. (Functions I i has_derivative f' x) (at x))"
surj_def: fixes f :: "'b \<Rightarrow> 'a" shows "top_pres (\<P> f) = (\<forall>y. \<exists>x. y = f x)"
nat_term_order_eqD: fixes to1 :: "'a nat_term_order"   and to2 :: "'a nat_term_order"   and dg :: "bool"   and ps :: "bool"   and u :: "'a"   and v :: "'a" assumes "nat_term_order_eq to1 to2 dg ps"   and "dg \<Longrightarrow> deg_pp (fst (rep_nat_term u)) = deg_pp (fst (rep_nat_term v))"   and "ps \<Longrightarrow> snd (rep_nat_term u) = snd (rep_nat_term v)" shows "nat_term_compare to1 u v = nat_term_compare to2 u v"
eval_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b" assumes "cartesian_closed_category C"   and "partial_magma.ide C b"   and "partial_magma.ide C c" shows "partial_magma.in_hom C (cartesian_closed_category.eval C b c) (elementary_category_with_binary_products.prod C (category_with_binary_products.pr0 C) (category_with_binary_products.pr1 C) (cartesian_closed_category.exp C b c) b) c"