hcju/isabelleps huggingface dataset:
wls_Abs_swap_all: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and x' :: "'var" assumes "wls s X"   and "wls s X'" shows "(Abs xs x X = Abs xs x' X') = (\<forall>y. (y = x \<or> fresh xs y X) \<and> (y = x' \<or> fresh xs y X') \<longrightarrow> X #[y \<and> x]_xs = X' #[y \<and> x']_xs)"
Rp_2: shows "LINT x|M. fp x * real_transfer_operator g x = LINT x|M. fp (T x) * g x"
widen_trans: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "subtype P S U"   and "subtype P U T" shows "subtype P S T"
ccInf_empty: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf {} = top"
dim_UNIV: shows "tangent_space.dim p (tangent_space p) = card basis"
assms_3: shows "(\<Prod>i\<in>#M. i) = (\<Prod>i\<in>#N. i)"
complex_roots_int_2: fixes p :: "int poly" shows "length (complex_roots_int p) = degree p"
TensorDiag_assoc: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t"   and "monoidal_language.Diag C u"   and "monoidal_language.Diag C v" shows "(t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> v = t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> v"
c: shows "(\<Sum>i\<in>UNIV. c i *s row i A) = 0"
bind_UNION: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'b set" shows "A \<bind> f = \<Union> (f ` A)"
a: shows "mopup_jump_over1 (Suc (2 * n), l, []) lm n ires"
matches_add_match_no_matching_Goto_simp2: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and rs :: "'c rule list" assumes "matches \<gamma> m p"   and "no_matching_Goto \<gamma> p rs" shows "no_matching_Goto \<gamma> p (add_match m rs)"
inject_10: fixes x10 :: "'c"   and y10 :: "'c" shows "(call(x10) = call(y10)) = (x10 = y10)"
eq: shows "concat (Ref \<C> us) = concat (Dec \<C> concat us)"
coeff_int_signof_mult: fixes x :: "nat \<Rightarrow> nat"   and f :: "'b poly"   and i :: "int" shows "coeff_int (signof x * f) i = signof x * coeff_int f i"
assms_2: shows "(x, e) \<in> set \<Gamma>"
reduce_aform_refine: fixes xi :: "(real \<times> real pdevs) list"   and x :: "real list set"   and d :: "nat"   and prec :: "nat"   and C :: "(real \<times> real pdevs) list \<Rightarrow> nat \<Rightarrow> real list \<Rightarrow> bool"   and r :: "unit" assumes "(xi, x) \<in> aforms_rel"   and "length xi = d" shows "(RETURN (reduce_aforms prec C xi), reduce_specs d r x) \<in> \<langle>aforms_rel\<rangle>nres_rel"
interval_lebesgue_integral_add_2: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b" assumes "interval_lebesgue_integrable M a b f"   and "interval_lebesgue_integrable M a b g" shows "LINT x=a..b|M. (f x + g x) = interval_lebesgue_integral M a b f + interval_lebesgue_integral M a b g"
sprefix_append: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c stream" shows "sprefix (xs @ ys) zs = (sprefix xs zs \<and> sprefix ys (sdrop (length xs) zs))"
tauLaw3: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> \<alpha>\<cdot>(\<tau>.P) \<approx> \<alpha>\<cdot>P"
gT_2: shows "wf\<^sub>t\<^sub>r\<^sub>m (Fun g T)"
maximum: shows "PRE (\<lambda>s. True) (IF (\<lambda>s. s ''y'' \<le> s ''x'') THEN (''z'' ::= (\<lambda>s. s ''x'')) ELSE (''z'' ::= (\<lambda>s. s ''y'')) FI) POST (\<lambda>s. s ''z'' = max (s ''x'') (s ''y''))"
stat_overrides_commonD: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides\<^sub>S old" shows "declclass new \<noteq> Object \<and> \<not> is_static new \<and> msig new = msig old \<and> G\<turnstile>declclass new\<prec>\<^sub>C declclass old \<and> G\<turnstile>Method new declared_in declclass new \<and> G\<turnstile>Method old declared_in declclass old"
nlists_alt_def: fixes A :: "'c set"   and n :: "nat" shows "nlists A n = {xs. set xs \<subseteq> A \<and> length xs = n}"
default_secure_IFS: fixes G :: "'v graph"   and f :: "('v \<times> 'v) set"   and nP :: "'v \<Rightarrow> 'a" assumes "wf_graph G"   and "f \<in> set_offending_flows G nP" shows "\<forall>i\<in>snd ` f. \<not> sinvar G (nP(i := \<bottom>))"
nth_root_nat_less: fixes k :: "nat"   and n :: "nat"   and x :: "nat" assumes "0 < k"   and "n < x ^ k" shows "nth_root_nat k n < x"
pow_equals: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat"   and n'' :: "nat" assumes "PoW t n = PoW t n'"   and "n \<le> n'"   and "n \<le> n''"   and "n'' \<le> n'" shows "PoW t n = PoW t n''"
that: fixes x :: "Syntax.var"   and e :: "Syntax.term" assumes "v = Lam x e \<or> v = Rec x e"   and "atom x \<sharp> c" shows "thesis"
vs: shows "vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t)"
card_face_cycle_sets_da0: shows "card H.face_cycle_sets = card G.face_cycle_sets - card {G.face_cycle_set a, G.face_cycle_set a'} + card (H.face_cycle_set ` (G.face_cycle_set a \<union> G.face_cycle_set a' - {a, a'}))"
length_Cons: fixes x :: "'f"   and xs :: "'f list" shows "length (x # xs) = Suc (length xs)"
path_strategies_strategy: fixes P :: "'a llist"   and n :: "nat" assumes "lset P \<subseteq> S - W"   and "enat n < llength P" shows "strategy p (path_strategies P $ n)"
cblinfun_image_mono: fixes S :: "'a ccsubspace"   and T :: "'a ccsubspace"   and A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "S \<le> T" shows "A *\<^sub>S S \<le> A *\<^sub>S T"
corthogonal_def: fixes vs :: "'a vec list" shows "corthogonal vs \<equiv> \<forall>i<length vs. \<forall>j<length vs. (inner_prod (vs ! j) (vs ! i) = (0::'a)) = (i \<noteq> j)"
ticks_image_sub: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and c :: "clock" assumes "dilating f sub r"   and "hamlet (Rep_run r n c)" shows "\<exists>n\<^sub>0. f n\<^sub>0 = n"
commut_normal: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and N :: "'c set" assumes "Group.group G"   and "subgroup H G"   and "N \<lhd> G" shows "H <#>\<^bsub>G\<^esub> N = N <#>\<^bsub>G\<^esub> H"
insert_Diff1: fixes x :: "'b"   and B :: "'b set"   and A :: "'b set" assumes "x \<in> B" shows "insert x A - B = A - B"
assms_1: shows "HT_partial_mods \<pi> mods P c Q"
a: shows "T\<^sub>p_opt init qs = T\<^sub>p init qs Strat"
substT_pls: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "'b"   and t2 :: "'b"   and t :: "'b"   and x :: "'a" assumes "Syntax_Arith_aux var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "t \<in> trm"   and "x \<in> var" shows "substT (pls t1 t2) t x = pls (substT t1 t x) (substT t2 t x)"
assms_1: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"
coeff_keys: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'c mpoly" shows "(m \<in> keys (mapping_of p)) = (coeff p m \<noteq> (0::'c))"
PUadmit_def: fixes \<sigma> :: "('a, 'b, 'c) subst"   and \<theta> :: "('a, 'b, 'c) hp"   and U :: "('c + 'c) set" shows "PUadmit \<sigma> \<theta> U = (\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGP \<theta>)) \<inter> U = {})"
iarray_to_vec_vec_to_iarray: fixes A :: "('b, 'c) vec" shows "iarray_to_vec (vec_to_iarray A) = A"
ex_conv_ex_lookup: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('b, 'c) mapping_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Mapping2.ex P t = (\<exists>k v. RBT_Mapping2.lookup t k = Some v \<and> P k v)"
card_subset_finite: fixes X :: "'a set"   and Y :: "'a set"   and n :: "nat" assumes "finite X"   and "card Y = n"   and "Y \<subseteq> X" shows "n \<le> card X"
axioms_1: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" assumes "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" shows "presheaf_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
vsubset_vdiff_vinsert: fixes A :: "V"   and B :: "V"   and x :: "V"   and C :: "V" shows "(A \<subseteq>\<^sub>\<circ> B -\<^sub>\<circ> vinsert x C) = (A \<subseteq>\<^sub>\<circ> B -\<^sub>\<circ> C \<and> x \<notin>\<^sub>\<circ> A)"
H: shows "H \<in> [{..<M}]\<^bsup>(\<alpha> ! i)\<^esup>"
False: shows "component_of_term (lt p) \<noteq> component_of_term (lt q)"
pr_strict: fixes fn :: "'f \<times> nat"   and gm :: "'f \<times> nat" shows "pr_strict fn gm = (pr_weak fn gm \<and> \<not> pr_weak gm fn)"
assms_4: shows "liminf (\<lambda>x. ereal (real_of_int (a x) / real_of_int (b x))) = 0"
frechet_blin: fixes I :: "('sf, 'sc, 'sz) interp"   and \<theta> :: "('sf, 'sz) trm" assumes "is_interp I"   and "dfree \<theta>" shows "(\<lambda>v. Blinfun (frechet I \<theta> v)) = blin_frechet (good_interp I) (simple_term \<theta>)"
elims_2: fixes x :: "'c"   and xa :: "'c edge list"   and xb :: "'c" assumes "Graph.ces x xa xb" obtains v1 :: "'c" and v2 :: "'c" where "x = v1__" and "xa = []" and "xb = v2__" and "v1__ = v2__"   | v1 :: "'c" and e :: "'c edge" and es :: "'c edge list" and v2 :: "'c" where "x = v1" and "xa = e # es" and "xb = v2" and "src e = v1 \<and> Graph.ces (tgt e) es v2"
arr_cod: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "arr (cod f)"
divide_less_eq: 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 less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide b c) a = (if less zero c then less b (times a c) else if less c zero then less (times a c) b else less zero a)"
Par_strict_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D \<or> B A ParStrict C D \<or> A B ParStrict D C \<or> B A ParStrict D C \<or> C D ParStrict A B \<or> C D ParStrict B A \<or> D C ParStrict A B \<or> D C ParStrict B A" shows "A B ParStrict C D"
canI_associator_1_1: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<a>[trg f, f, g] = can (\<^bold>\<langle>trg f\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>) ((\<^bold>\<langle>trg f\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>)"
add_less_imp_less_left: fixes c :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" assumes "c + a \<subset># c + b" shows "a \<subset># b"
igSwapIGAbs_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSwapIGAbs MOD \<equiv> \<forall>zs z1 z2 xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igSwapAbs MOD zs z1 z2 (igAbs MOD xs x X) = igAbs MOD xs (x @xs[z1 \<and> z2]_zs) (igSwap MOD zs z1 z2 X)"
cenum_all_UNIV: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "cEnum = Some (enum, enum_all, enum_ex)" shows "enum_all P = Ball UNIV P"
is_Insert_def: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" shows "is_Insert stateful_strand_step = (\<exists>x41 x42. stateful_strand_step = insert\<langle>x41,x42\<rangle>)"
inject_2: fixes x21 :: "variable"   and x22 :: "String.literal set"   and y21 :: "variable"   and y22 :: "String.literal set" shows "(Tv x21 x22 = Tv y21 y22) = (x21 = y21 \<and> x22 = y22)"
assms_2: fixes x :: "'a" assumes "x \<notin> A" shows "f x = dflt"
pderiv_singleton: fixes a :: "'b" shows "pderiv [:a:] = 0"
eval_red2_Diag_Unity: fixes a :: "'c local.term" assumes "Ide a"   and "Diag a" shows "\<lbrace>a \<^bold>\<Down> \<^bold>\<I>\<rbrace> = \<r>[\<lbrace>a\<rbrace>]"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Top of Top \<Rightarrow> f1 | Bot \<Rightarrow> f2) = f1"
add_commute: fixes a :: "'a ppath set"   and b :: "'a ppath set" shows "a \<union> b = b \<union> a"
prems_3: shows "(N, \<alpha>) \<in> \<RR>"
mtf2_forward_beforeq: fixes q :: "'a"   and xs :: "'a list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "i < index xs q" shows "index (mtf2 n q xs) (xs ! i) \<le> index xs q"
select_convs_3: fixes states :: "'a \<Rightarrow> 'b"   and msgs :: "nat \<Rightarrow> 'c message list"   and process_snapshot :: "'a \<Rightarrow> 'b option"   and channel_snapshot :: "nat \<Rightarrow> 'c list \<times> recording_state"   and more :: "'d" shows "ps \<lparr>states = states, msgs = msgs, process_snapshot = process_snapshot, channel_snapshot = channel_snapshot, \<dots> = more\<rparr> = process_snapshot"
simps_4: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s0 :: "'a"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and realistic_execution :: "'b list list \<Rightarrow> bool"   and control :: "'a \<Rightarrow> 'd \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and AS_set :: "'b list set"   and invariant :: "'a \<Rightarrow> bool"   and AS_precondition :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and aborting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and waiting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and interrupt :: "nat \<Rightarrow> bool"   and n :: "nat"   and execs :: "'d \<Rightarrow> 'b list list"   and s :: "'a" assumes "Interruptible_Separation_Kernel kstep output_f s0 current cswitch kprecondition realistic_execution control kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting"   and "\<not> interrupt (Suc n)"   and "\<not> (execs (current s) = [] \<or> execs (current s) = [[]])" shows "Interruptible_Separation_Kernel.run_total kstep current cswitch interrupt control (Suc n) s execs = Interruptible_Separation_Kernel.run_total kstep current cswitch interrupt control n (Kernel.step kstep (Kernel.next_state current control s execs) (Kernel.next_action current control s execs)) (Kernel.next_execs current control s execs)"
plus_left_id: fixes x :: "int poly fract" shows "rat_poly_plus 0 x = x"
loop_super_orientable_def: fixes x :: "'a" shows "loop_super_orientable x \<equiv> \<exists>y. loop_super_orientation x y"
sup2: fixes B :: "'b"   and k :: "'b"   and b :: "'b"   and a :: "'b" assumes "B \<equiv> sup k b" shows "sup a B \<equiv> sup k (sup a b)"
Knat_2: shows "expectation (\<lambda>x. \<bar>u 1 x\<bar> * indicat_real {x \<in> space M. real Knat \<le> \<bar>u 1 x\<bar>} x) < rho"
uint32_shiftr_def: fixes x :: "uint32"   and n :: "integer" shows "uint32_shiftr x n = (if n < 0 \<or> 32 \<le> n then undefined drop_bit x n else drop_bit (nat_of_integer n) x)"
Least_le: fixes P :: "'d \<Rightarrow> bool"   and k :: "'d" assumes "P k" shows "(LEAST x. P x) \<le> k"
A6ch: shows "data15 \<in> OUT sA6 \<and> stA6 \<in> VAR sA6"
eventually_poly_nz_at_within: fixes p :: "'a poly"   and x :: "'a"   and S :: "'a set" assumes "p \<noteq> 0" shows "\<forall>\<^sub>F x in at x within S. poly p x \<noteq> (0::'a)"
split_set_path_length: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and n :: "nat" shows "{Cs. Subobjs P C Cs \<and> length Cs \<le> Suc n} = {Cs. Subobjs P C Cs \<and> length Cs \<le> n} \<union> {Cs. Subobjs P C Cs \<and> length Cs = Suc n}"
ass_4: fixes k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "(k, \<gamma>) \<in> set (a # l)" shows "valid_path \<gamma>"
abelian_subgroup_generated: fixes G :: "('a, 'b) monoid_scheme"   and S :: "'a set" assumes "Group.group G"   and "comm_group G" shows "comm_group (subgroup_generated G S)"
wf_qorder_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.wf_qorder less_eq less \<equiv> well_founded_ordering UNIV less_eq less"
index_map_vec_1: fixes i :: "nat"   and v :: "'a vec"   and f :: "'a \<Rightarrow> 'b" assumes "i < dim_vec v" shows "map_vec f v $ i = f (v $ i)"
distinct_8: fixes x5 :: "('a, 'b, 'c) expr"   and x1 :: "('a, 'b, 'c) val" shows "Read x5 \<noteq> VE x1"
subst_comp_eq_if_disjoint_vars: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<gamma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> (subst_domain \<gamma> \<union> range_vars \<gamma>) = {}" shows "\<gamma> \<circ>\<^sub>s \<delta> = \<delta> \<circ>\<^sub>s \<gamma>"
Red_F_Sup_in_Liminf: shows "Red_F (Sup_llist Ns) \<subseteq> Red_F (Liminf_llist Ns)"
transfer_rule_numeral: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "R (0::'a) (0::'b)"   and "R (1::'a) (1::'b)"   and "(R ===> R ===> R) (+) (+)" shows "((=) ===> R) numeral numeral"
matrix_inv_right: fixes M :: "(('a, 'b) vec, 'c) vec" assumes "invertible M" shows "M ** matrix_inv M = mat (1::'a)"
map_def: fixes f :: "'a" shows "\<tau>.A.map f = (if \<tau>.A.arr f then f else \<tau>.A.null)"
adjoint_idem: fixes f :: "'a set \<Rightarrow> 'a set" shows "adjoint (adjoint f) = f"
iso_is_retraction: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "local.iso f" shows "retraction f"
group_comm_groupI: assumes "\<And>x y. \<lbrakk>x \<in> carrier (mat_monoid nr nc b); y \<in> carrier (mat_monoid nr nc b)\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>mat_monoid nr nc b\<^esub> y = y \<otimes>\<^bsub>mat_monoid nr nc b\<^esub> x" shows "comm_group (mat_monoid nr nc b)"
is_processT3: fixes P :: "'b process" shows "\<forall>s t. (s @ t, {}) \<in> F P \<longrightarrow> (s, {}) \<in> F P"
not_bad: shows "Sigma_Algebra.measure (measure_spmf (map_spmf (\<lambda>x. (f x, bad x)) p)) (A \<times> {False}) = Sigma_Algebra.measure (measure_spmf (map_spmf (\<lambda>x. (f x, bad x)) q)) (A \<times> {False})"
greatest_less_zero_row: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c" assumes "reduced_row_echelon_form_upt_k A k"   and "is_zero_row_upt_k i k A"   and "\<not> (\<forall>a. is_zero_row_upt_k a k A)" shows "(GREATEST m. \<not> is_zero_row_upt_k m k A) < i"
H_cons: fixes p :: "'a"   and p' :: "'a"   and q' :: "'a"   and q :: "'a"   and x :: "'a" assumes "d p \<le> d p'"   and "d q' \<le> d q"   and "H p' x q'" shows "H p x q"
IH_4: assumes "insert\<langle>t,s\<rangle> \<in> set S" shows "fv t \<union> fv s \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
assms_2: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> x"
rsp: shows "eq_onp (\<lambda>x. x \<in> {(a, b). a \<le> b}) (0::'a, 0::'a) (0::'a, 0::'a)"
strong_induct: fixes x1 :: "(W.var \<times> tyS) list"   and x2 :: "trm"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (W.var \<times> tyS) list \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes " x1 \<turnstile> x2 : x3 "   and "\<And>\<Gamma> x S T y. \<lbrakk>valid \<Gamma>; (x, S) \<in> set \<Gamma>; T \<prec> S\<rbrakk> \<Longrightarrow> P y \<Gamma> (Var x) T"   and "\<And>\<Gamma> t\<^sub>1 T\<^sub>1 T\<^sub>2 t\<^sub>2 y. \<lbrakk> \<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1\<rightarrow>T\<^sub>2 ; \<And>z. P z \<Gamma> t\<^sub>1 (T\<^sub>1\<rightarrow>T\<^sub>2);  \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1 ; \<And>z. P z \<Gamma> t\<^sub>2 T\<^sub>1\<rbrakk> \<Longrightarrow> P y \<Gamma> (App t\<^sub>1 t\<^sub>2) T\<^sub>2"   and "\<And>x \<Gamma> T\<^sub>1 t T\<^sub>2 y. \<lbrakk>x \<sharp> \<Gamma>;  (x, Ty T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2 ; \<And>z. P z ((x, Ty T\<^sub>1) # \<Gamma>) t T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> (Lam [x].t) (T\<^sub>1\<rightarrow>T\<^sub>2)"   and "\<And>x \<Gamma> t\<^sub>1 T\<^sub>1 t\<^sub>2 T\<^sub>2 y. \<lbrakk>(ftv T\<^sub>1 - ftv \<Gamma>) \<sharp>* y; x \<sharp> \<Gamma>;  \<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1 ; \<And>z. P z \<Gamma> t\<^sub>1 T\<^sub>1;  (x, close \<Gamma> T\<^sub>1) # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2 ; \<And>z. P z ((x, close \<Gamma> T\<^sub>1) # \<Gamma>) t\<^sub>2 T\<^sub>2; (ftv T\<^sub>1 - ftv \<Gamma>) \<sharp>* T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> (Let x be t\<^sub>1 in t\<^sub>2) T\<^sub>2" shows "P z x1 x2 x3"
simps_1: fixes pending_write :: "bool"   and A :: "nat set" shows "acquired pending_write [] A = (if pending_write then A else {})"
other_def: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "'e set"   and \<sigma> :: "'e \<Rightarrow> 'd"   and \<sigma>' :: "'e \<Rightarrow> 'd" shows "other P I \<sigma> \<sigma>' \<equiv> \<forall>i. if i \<in> I then \<sigma>' i = \<sigma> i else P (\<sigma> i) (\<sigma>' i)"
axioms_1: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "NFlow c s t f" shows "NPreflow c s t f"