hcju/isabelleps huggingface dataset:
l5_5_1: 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 "Tarski_neutral_dimensionless.Le Bet Cong A B C D" shows "\<exists>X. Bet A B X \<and> Cong A X C D"
HSem_def': fixes \<Gamma> :: "('a \<times> 'e) list"   and \<rho> :: "'a \<Rightarrow> 'b" shows "HSem1.HSem_syn \<Gamma> \<rho> = (\<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> HSem1.EvalHeapSem_syn \<Gamma> x)"
det_echelon_form: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "echelon_form A" shows "det A = (\<Prod>i\<in>UNIV. A $ i $ i)"
Mapping_lookup_update: fixes k :: "'b"   and v :: "'a"   and m :: "('b, 'a) mapping"   and k' :: "'b" shows "Mapping.lookup (Mapping.update k v m) k' = (if k = k' then Some v else Mapping.lookup m k')"
coeff_in_ring: fixes a :: "'e \<Rightarrow> 'a"   and A :: "'e set"   and x :: "'e" assumes "a \<in> A \<rightarrow> carrier R"   and "x \<in> A" shows "a x \<in> carrier R"
cases: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" obtains "x = []"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" where "x = [t__]"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # t' # bq"
M_supset: shows "timpl_closure_set M T \<subseteq> M"
that_8: fixes f :: "Value \<rightarrow> Value"   and b\<^sub>2 :: "bool" assumes "x = Fn\<cdot>f"   and "y = CB\<cdot>(Discr b\<^sub>2)" shows "thesis"
of_nat_0_less_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (semiring_1.of_nat one plus zero n) = (0 < n)"
list_map_to_fun_create_map: fixes ias :: "('a \<times> 'b) list"   and i :: "'a" shows "set (list_map_to_fun (create_map ias) i) = set ias \<inter> {i} \<times> UNIV"
boundedE: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b \<sqinter> c \<le> a" obtains "b \<le> a" and "c \<le> a"
sincos_principal_value: fixes x :: "real" shows "\<exists>y. (- pi < y \<and> y \<le> pi) \<and> sin y = sin x \<and> cos y = cos x"
Step1_4_4b: fixes p :: "PrIds"   and l :: "Locs"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> Write rmCh mm ires p l \<and> $S4 rmhist p \<and> S4 rmhist p$ \<and> unchanged (e p, c p, r p, rmhist!p) \<longrightarrow> Write memCh mm (resbar rmhist) p l"
facts_2: fixes n :: "nat" assumes "2 * card (uverts H) \<le> n" shows "prob_space_with_indicators (MGn p n) {S. S \<subseteq> {1..n} \<and> card S = card (uverts H)} (\<lambda>S. {es \<in> space (MGn p n). subgraph (f S) (induced_subgraph S (edge_space.edge_ugraph n es))})"
less_le_not_le: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.preorder less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
disjointed_subset: fixes A :: "nat \<Rightarrow> 'c set"   and n :: "nat" shows "disjointed A n \<subseteq> A n"
list_assn_pure: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "is_pure P" shows "is_pure (list_assn P)"
fds_truncate_inverse: fixes m :: "nat"   and f :: "'b fds" shows "fds_truncate m (inverse (fds_truncate m f)) = fds_truncate m (inverse f)"
set_intros_1: fixes x3 :: "'b" shows "x3 \<in> atoms_ltlc prop\<^sub>c(x3)"
measurable_compose: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and g :: "'e \<Rightarrow> 'f"   and L :: "'f measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L" shows "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M L"
GB_Macaulay_Dube_def: fixes fs :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list" shows "GB_Macaulay_Dube fs = punit.Macaulay_list (deg_shifts_pp (Indets_pp fs) (Dube (Suc (length (Indets_pp fs))) (max_list (map poly_deg_pp fs))) fs)"
FSwSbImorph_iff: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "'g \<Rightarrow> 'h"   and hA :: "'i \<Rightarrow> 'j"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'k, 'g, 'i, 'l) model_scheme"   and MOD' :: "('b, 'c, 'd, 'e, 'f, 'k, 'h, 'j, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.FSwSbImorph Delta h hA MOD MOD' = (FixSyn.FSbImorph Delta h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD')"
subsFConj_def: fixes atms :: "(signs \<times> predicate \<times> vbl list) list"   and nAs :: "(nat \<times> formula) list"   and z :: "signs"   and A0 :: "formula"   and A1 :: "formula" shows "subsFConj atms nAs z A0 A1 = (case z of Pos \<Rightarrow> {(atms, (0, A0) # nAs), (atms, (0, A1) # nAs)} | Neg \<Rightarrow> {(atms, (0, A0) # (0, A1) # nAs)})"
abs_minus: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'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 a :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (uminus a) = abs a"
cases: fixes a1 :: "(Location \<times> Location) set"   and a2 :: "Val"   and a3 :: "Val" assumes "(a1, a2, a3) \<in> twiddleVal" obtains \<beta> :: "(Location \<times> Location) set" where "a1 = \<beta>'__" and "a2 = RVal Nullref" and "a3 = RVal Nullref"   | l1 :: "Location" and l2 :: "Location" and \<beta> :: "(Location \<times> Location) set" where "a1 = \<beta>''__" and "a2 = RVal (Loc l1)" and "a3 = RVal (Loc l2)" and "(l1, l2) \<in> \<beta>''__"   | i1 :: "int" and i2 :: "int" and \<beta> :: "(Location \<times> Location) set" where "a1 = \<beta>" and "a2 = IVal i1" and "a3 = IVal i2" and "i1 = i2"
sel: fixes x1 :: "'d" shows "the_LabelN ln x1 = x1"
aadd_pos_le: fixes a :: "ant"   and b :: "ant" assumes "0 \<le> a" shows "b \<le> a + b"
lc: shows "lead_coeff p * lead_coeff q \<noteq> (0::'a)"
intro: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleHom G smult V smult' T"   and "FGModuleIso_axioms V T W" shows "FGModuleIso G smult V smult' T W"
ith_def: fixes v :: "(nat \<Rightarrow> real) \<times> nat"   and i :: "nat" shows "v\<^bsub>i\<^esub> = fst v i"
Cod_ide: fixes f :: "('a E.term, 'a) arr" assumes "ide f" shows "Cod f = Dom f"
pos_1: fixes k :: "nat" assumes "k < m" shows "0 < d fs' k"
sylvester_vec_poly: fixes v :: "'b vec"   and p :: "'b poly"   and q :: "'b poly" assumes "v \<in> carrier_vec (degree p + degree q)" shows "poly_of_vec ((sylvester_mat p q)\<^sup>T *\<^sub>v v) = poly_of_vec (vec_first v (degree q)) * p + poly_of_vec (vec_last v (degree p)) * q"
IH: fixes ys :: "'a list"   and zs :: "'a list" assumes "map f xs' = ys @ zs" shows "\<exists>vs ws. xs' = vs @ ws \<and> map f vs = ys \<and> map f ws = zs"
iterates_conv_inf_llist: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "iterates f a = inf_llist (\<lambda>n. (f ^^ n) a)"
frac_less2: fixes x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "(0::'a) < x"   and "x \<le> y"   and "(0::'a) < w"   and "w < z" shows "x / z < y / w"
ide_def: fixes a :: "('o, 'a) arr" shows "ide a \<equiv> COMP a a \<noteq> null \<and> (\<forall>f. (COMP f a \<noteq> null \<longrightarrow> COMP f a = f) \<and> (COMP a f \<noteq> null \<longrightarrow> COMP a f = f))"
create_character_data_def: fixes document_ptr :: "(_) document_ptr"   and "text" :: "char list" shows "create_character_data document_ptr text = Heap_Error_Monad.bind new_character_data (\<lambda>new_character_data_ptr. Heap_Error_Monad.bind (set_val new_character_data_ptr text) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes document_ptr (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr # disc_nodes)) (\<lambda>_. return new_character_data_ptr))))"
lattice_ow_def: fixes U :: "'c set"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ls :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "lattice_ow U inf le ls sup \<equiv> semilattice_inf_ow U inf le ls \<and> semilattice_sup_ow U sup le ls"
mult_coprime: fixes a :: "nat"   and b :: "nat" assumes "coprime a b" shows "f (a * b) = f a * f b"
c3: assumes "sqn rt ip = \<pi>\<^sub>2 r"   and "\<pi>\<^sub>5 r < the (dhops rt ip)" shows "P (rt(ip \<mapsto> r))"
left_neutral: fixes a :: "'d upred" shows "(false \<or> a) = a"
assms_2: shows "{0..<length xs} \<subseteq> A \<union> B"
emp_not_in_code: shows "\<epsilon> \<notin> \<C>"
F_def: fixes i :: "nat" shows "F i = (if i \<in> J then F' i else space (M i))"
sorted_quicksort_by_rel: fixes xs :: "'a list" shows "sorted (quicksort_by_rel (\<le>) [] xs)"
assms: shows "P \<Longrightarrow>\<^sub>l\<^sup>^Rs"
power_even_eq: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (2 * n) = power.power one times (power.power one times a n) 2"
Abs_cfun_eqvt: fixes f :: "'a \<Rightarrow> 'b"   and p :: "perm" assumes "cont f" shows "(p \<bullet> Abs_cfun) f = Abs_cfun f"
chainI: fixes Y :: "nat \<Rightarrow> 'a" assumes "\<And>i. Y i \<sqsubseteq> Y (Suc i)" shows "chain Y"
dM1: shows "M1 \<in> carrier_mat d1 d1"
List_unique1: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and as :: "ref list" assumes "List p h as" shows "\<exists>!as. List p h as"
rotate1_length01: fixes xs :: "'a list" assumes "length xs \<le> 1" shows "rotate1 xs = xs"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "qtname \<Rightarrow> 'a"   and f2 :: "ty \<Rightarrow> int \<Rightarrow> 'a"   and obj_tag :: "obj_tag" shows "P (case obj_tag of CInst x \<Rightarrow> f1 x | Arr x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. obj_tag = CInst x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. obj_tag = Arr x21 x22 \<longrightarrow> P (f2 x21 x22)))"
joinablep_def: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" shows "joinablep P x y = (\<exists>z. P x z \<and> P y z)"
RID\<^sub>SI: fixes \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option"   and l :: "'e"   and v :: "('d, 'e, 'f) val"   and r :: "'d" assumes "\<sigma> l = Some v"   and "r \<in> RID\<^sub>V v" shows "r \<in> RID\<^sub>S \<sigma>"
simps_3: shows "extended_to_set \<stileturn> = {}"
mono_ExcessTab: fixes g' :: "graph"   and p :: "nat"   and g :: "graph" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "Invariants.inv g" shows "set (ExcessTable g (vertices g)) \<subseteq> set (ExcessTable g' (vertices g'))"
lmult_0: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and m :: "'j" assumes "module R M"   and "m \<in> carrier M" shows "\<zero>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> m = \<zero>\<^bsub>M\<^esub>"
assms_1: shows "f \<in> quasi_measure_preserving M N"
left_idem: fixes a :: "'a"   and b :: "'a" shows "gcd a (gcd a b) = gcd a b"
ctr_2: fixes xs :: "'c llist"   and ys :: "'d llist" assumes "\<not> lnull xs"   and "\<not> lnull ys" shows "lzip xs ys = LCons (lhd xs, lhd ys) (lzip (ltl xs) (ltl ys))"
dim_unique: fixes B :: "'a set"   and V :: "'a set"   and n :: "nat" assumes "B \<subseteq> UNIV"   and "V \<subseteq> UNIV"   and "B \<subseteq> V"   and "V \<subseteq> span B"   and "independent B"   and "card B = n" shows "vs1.dim V = n"
lookup_filter: fixes P :: "'c \<times> 'b \<Rightarrow> bool"   and xs :: "('c, 'b) oalist_ntm"   and k :: "'c" shows "OAlist_lookup_ntm (OAlist_filter_ntm P xs) k = (let v = OAlist_lookup_ntm xs k in if P (k, v) then v else (0::'b))"
lprefix_lnthD: fixes xs :: "'b llist"   and ys :: "'b llist"   and n :: "nat" assumes "lprefix xs ys"   and "enat n < llength xs" shows "xs ?! n = ys ?! n"
mk_poly_dim_1: fixes A :: "'b mat" shows "dim_col (mk_poly A) = dim_col A"
diff_eq_singleton_imp: fixes M :: "'a multiset"   and A :: "'a set"   and a :: "'a" assumes "M -s A = {#a#}" shows "a \<in> set_mset M - A"
order_gt_1_gen_not_1: fixes G :: "('d, 'e) cyclic_group_scheme" assumes "Cyclic_Group.cyclic_group G"   and "1 < order G" shows "\<^bold>g\<^bsub>G\<^esub> \<noteq> \<one>\<^bsub>G\<^esub>"
le_less: fixes y :: "'c"   and x :: "'c" shows "(y \<le> x) = (y < x \<or> x = y)"
mk_poly2_def: fixes A :: "'b mat" shows "mk_poly2 A \<equiv> mk_poly2_col A (0\<^sub>v (dim_row A)) (dim_col A)"
as0_7: shows "CAs0 \<cdot>\<cdot>cl \<eta>s0 = CAs"
app: fixes P :: "'a \<Rightarrow> bool"   and t :: "'a"   and u :: "'a" assumes "simple_syntactic_or P" shows "P (app t u) = (P t \<or> P u)"
index_is_2: shows "\<forall>i. i \<noteq> Suc 0 \<longrightarrow> i \<noteq> 3 \<longrightarrow> 0 < i \<longrightarrow> i < 4 \<longrightarrow> i = 2"
avl_fib_bound_auto: fixes t :: "('b \<times> nat) tree" assumes "avl t" shows "fib (height t + 2) \<le> size1 t"
ennreal_approx_unit: fixes z :: "ennreal"   and y :: "ennreal" assumes "\<And>a. \<lbrakk>0 < a; a < 1\<rbrakk> \<Longrightarrow> a * z \<le> y" shows "z \<le> y"
slo_defs_3: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
l_cancel_one: fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier \<G>"   and "a \<in> carrier \<G>" shows "(x \<otimes> a = x) = (a = \<one>)"
val_in_P_valuation: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "v' \<in> P\<^bsub> K v\<^esub>" shows "valuation K v'"
UNION_disjoint: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "sum g (\<Union> (A ` I)) = (\<Sum>x\<in>I. sum g (A x))"
mrconc: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and x :: "'c"   and p :: "'b list"   and a :: "'b"   and b :: "'b" assumes "applied_rule_rev C x p = \<lfloor>a\<rfloor>" shows "applied_rule_rev C x (b # p) = \<lfloor>a\<rfloor>"
SystemProof_def: shows "SystemProof = (SystemConcl, [(0, Rrule ImplyR 0), (0, Lrule AndL 0), (0, Cut SystemDICut), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (0, Lrule ImplyL 0), (Suc (Suc 0), CloseId 0 0), (Suc 0, AxSubst ADIGeq SystemDISubst), (Suc 0, Rrule ImplyR 0), (Suc 0, CloseId 1 0), (0, Rrule ImplyR 0), (0, Cut SystemDCCut), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (0, Lrule EquivBackwardL 0), (0, Rrule CohideR 0), (0, AxSubst ADC SystemDCSubst), (0, CloseId 0 0), (0, Rrule CohideRR 0), (0, Cut SystemVCut), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (0, Cut SystemDECut), (0, Lrule EquivBackwardL 0), (0, Rrule CohideRR 0), (1, CloseId (Suc 1) 0), (Suc 1, CloseId 0 0), (1, AxSubst AV SystemVSubst), (0, Cut SystemVCut2), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (Suc 1, CloseId 0 0), (Suc 1, CloseId (Suc 2) 0), (Suc 1, AxSubst AV SystemVSubst2), (0, Rrule CohideRR 0), (0, DEAxiomSchema (OSing vid2 (Var vid1)) SystemDESubst), (0, Cut SystemKCut), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (0, AxSubst AK SystemKSubst), (0, CloseId 0 0), (0, Rrule CohideR 0), (1, AxSubst ADW SystemDWSubst), (0, G), (0, Cut SystemEquivCut), (0, Lrule EquivBackwardL 0), (0, Rrule CohideR 0), (0, CloseId 0 0), (0, Rrule CohideR 0), (0, CE SystemCEFml1 SystemCEFml2 SystemCESubst), (0, Rrule ImplyR 0), (0, Lrule AndL 0), (0, Cut SystemDiffAssignCut), (0, Lrule EquivBackwardL 0), (0, Rrule CohideRR 0), (0, CloseId 0 0), (0, CloseId 1 0), (0, AxSubst Adassign SystemDiffAssignSubst)])"
intro: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and one :: "'a" assumes "semiring_ow U plus times"   and "cancel_comm_monoid_add_ow U plus minus zero"   and "zero_neq_one_ow U one zero"   and "monoid_mult_ow U one times" shows "semiring_1_cancel_ow U plus minus zero one times"
Compl_eq_Diff_UNIV: fixes A :: "'b set" shows "- A = UNIV - A"
less_max_iff_disj: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "less z (ord.max less_eq x y) = (less z x \<or> less z y)"
set_append: fixes xs :: "'c list"   and ys :: "'c list" shows "set (xs @ ys) = set xs \<union> set ys"
mono_neutral_cong_left: fixes T :: "'a set"   and S :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. h i = (1::'b)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "prod g S = prod h T"
a_assoc: fixes x :: "rat vec"   and y :: "rat vec"   and z :: "rat vec" assumes "x \<in> Rn"   and "y \<in> Rn"   and "z \<in> Rn" shows "x + y + z = x + (y + z)"
Abs_cong: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and x' :: "'var" assumes "good X"   and "good X'"   and "fresh xs y X"   and "fresh xs y X'"   and "X #[y \<and> x]_xs = X' #[y \<and> x']_xs" shows "Abs xs x X = Abs xs x' X'"
dom: shows "\<D>\<^sub>\<circ> S = 3\<^sub>\<nat>"
arity_hd_eq_inf: fixes \<zeta> :: "('s, 'v) hd" shows "arity_hd \<zeta> = \<infinity>"
ring_mat_def: fixes ty :: "'b itself"   and n :: "nat"   and b :: "'c" shows "ring_mat ty n b \<equiv> \<lparr>carrier = carrier_mat n n, monoid.mult = (*), one = 1\<^sub>m n, zero = 0\<^sub>m n n, add = (+), \<dots> = b\<rparr>"
Domainp_cr_rat: shows "Domainp pcr_rat = (\<lambda>x. snd x \<noteq> 0)"
card_of_Fpow_infinite: fixes A :: "'b set" assumes "infinite A" shows "|Fpow A| =o |A|"
noz: shows "a \<noteq> 0 \<or> b \<noteq> 0 \<or> c \<noteq> 0"
smooth_compat_restrict_chartI2: fixes k :: "enat"   and c' :: "('c, 'd) chart"   and c :: "('c, 'd) chart"   and S :: "'c set" assumes "k-smooth_compat c' c" shows "k-smooth_compat c' (restrict_chart S c)"
devExt_greatest: fixes n' :: "nat"   and n'' :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "\<exists>n'''<n' + n''. \<not> Option.is_none (devBC t n nid n''')"   and "Option.is_none (devBC t n nid (n' + n''))"   and "n'' \<noteq> 0" shows "devExt t n nid n' n'' = bc (\<sigma>\<^bsub>the (devBC t n nid (GREATEST n'''. n''' < n' + n'' \<and> \<not> Option.is_none (devBC t n nid n''')))\<^esub>t (GREATEST n'''. n''' < n' + n'' \<and> \<not> Option.is_none (devBC t n nid n''')))"
Collect_conv_if2: fixes a :: "'d"   and P :: "'d \<Rightarrow> bool" shows "{x. a = x \<and> P x} = (if P a then {a} else {})"
map_of_map_keys: fixes xs :: "'a list"   and m :: "'a \<Rightarrow> 'b option" assumes "set xs = dom m" shows "map_of (map (\<lambda>k. (k, the (m k))) xs) = m"
l6: fixes x :: "'a" shows "- x \<sqinter> - - x = bot"
reparam_weak_def: fixes \<gamma>1 :: "real \<Rightarrow> 'a"   and \<gamma>2 :: "'b \<Rightarrow> 'a" shows "reparam_weak \<gamma>1 \<gamma>2 \<equiv> \<exists>\<phi>. (\<forall>x\<in>{0..1}. \<gamma>1 x = (\<gamma>2 \<circ> \<phi>) x) \<and> \<phi> piecewise_C1_differentiable_on {0..1} \<and> \<phi> 0 = (0::'b) \<and> \<phi> 1 = (1::'b) \<and> \<phi> ` {0..1} = {0::'b..1::'b}"
cobounded1: fixes a :: "'c"   and b :: "'c" shows "a \<le> a \<squnion> b"
carrierE: fixes x :: "'b" assumes "x \<in> sub.carrier" obtains ca :: "('b, 'b) chart" where "ca \<in> manifold_eucl.dest.charts_submanifold (codomain c)" and "x \<in> domain ca"
d_two_orders: fixes x :: "'a"   and y :: "'a" shows "(d x \<sqsubseteq> d y) = (d x \<le> d y)"
prems_9: shows "Terms.Let \<Gamma> body = Terms.Let \<Gamma>' body'"
nonzero_divide_mult_cancel_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "b \<noteq> zero" shows "divide b (times a b) = divide one a"
rte: fixes p :: "nat"   and i :: "nat" assumes "correct p (te p (i + 1))" shows "te p i \<le> te p (i + 1)"
simps_2: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and L :: "lock_action"   and Ls :: "lock_action list" shows "lock_actions_ok l t (L # Ls) = (lock_action_ok l t L \<and> lock_actions_ok (upd_lock l t L) t Ls)"
empty: fixes g :: "'d \<Rightarrow> 'c" shows "sum g {} = (0::'c)"
IH: fixes x2a :: "('fun, 'a) Term.term" assumes "x2a \<in> set T"   and "wf\<^sub>t\<^sub>r\<^sub>m (x2a \<cdot> \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m (x2a \<cdot> rm_vars X \<delta>)"
that: fixes document_ptr :: "(_) document_ptr" assumes "ptr = cast document_ptr" shows "thesis"
gap_le_Suc: fixes i :: "nat"   and t :: "nat" shows "gap i t \<le> gap i (Suc t)"
0: fixes f :: "'fsym" assumes "f \<in> fsymsL l" shows "intF1 f = intF2 f"
delete_map: fixes f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'c"   and k :: "'a"   and ps :: "('a \<times> 'b) list" assumes "\<And>kv. fst (f kv) = fst kv" shows "delete k (map f ps) = map f (delete k ps)"
distinct_pds_valuation1: fixes n :: "nat"   and j :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "0 < n"   and "j \<le> n"   and "distinct_pds K n P" shows "valuation K (\<nu>\<^bsub>K P j\<^esub>)"
prime_elem_int_not_dvd_neg1_power: fixes p :: "int"   and n :: "nat" assumes "prime_elem p" shows "\<not> p dvd (- 1) ^ n"
is_finite_def: fixes a :: "('a, 'b) IEEE.float" shows "is_finite a = (is_normal a \<or> is_denormal a \<or> is_zero a)"
asset_i_used: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "(Asset n, Crypt (Auth_ShaKey n) (PriKey A)) \<in> s \<longrightarrow> PriKey A \<in> used s"
extensional0_directional_derivative: fixes k :: "enat"   and a :: "'c"   and v :: "'c" shows "extensional0 (manifold_eucl.diff_fun_space k) (directional_derivative k a v)"
retraction_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b" assumes "category C" shows "category.retraction C g \<equiv> \<exists>f. partial_magma.ide C (C g f)"
of_rat_rat: fixes b :: "int"   and a :: "int" assumes "b \<noteq> 0" shows "of_rat (Rat.Fract a b) = of_int a / of_int b"
rtranclp_eq_rtranclp: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<lambda>x y. P x y \<or> x = y)\<^sup>*\<^sup>* = P\<^sup>*\<^sup>*"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 \<mu> (\<nu> \<star> \<pi>)"
i_F_Inf: shows "\<iota> \<in> F_Inf"
g_ode_inv_def: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and I :: "'b \<Rightarrow> bool" shows "x\<acute>=f & G on U S @ t\<^sub>0 DINV I  = x\<acute>=f & G on U S @ t\<^sub>0"