hcju/isabelleps huggingface dataset:
a2: shows " \<Gamma> \<subseteq> \<Gamma>' "
set_intros_1: fixes y :: "'c \<times> ('c, 'd) rose_tree\<^sub>m list"   and x :: "('c \<times> ('c, 'd) rose_tree\<^sub>m list, 'd \<times> 'd rose_tree\<^sub>h list) blindable\<^sub>m"   and ya :: "'c" assumes "y \<in> set1_blindable\<^sub>m x"   and "ya \<in> fsts y" shows "ya \<in> set1_rose_tree\<^sub>m (Tree\<^sub>m x)"
imp: fixes f :: "'a \<Rightarrow> 'b"   and \<pi> :: "perm" assumes "cont f" shows "cont (\<pi> \<bullet> f)"
sbelow_sctns_def: fixes X :: "'b set"   and sctn :: "'b sctn set" shows "sbelow_sctns X sctn = SPEC (\<lambda>b. b \<longrightarrow> X \<subseteq> sbelow_halfspaces sctn)"
clause_entails_clause_def: fixes C :: "'b literal set"   and D :: "'b literal set" shows "clause_entails_clause C D = (\<forall>I. fo_interpretation I \<longrightarrow> validate_clause I C \<longrightarrow> validate_clause I D)"
Pd_singletons_for_ds_range: fixes X :: "'a set"   and ds :: "'b set"   and d :: "'b" shows "Field (Pd_singletons_for_ds X ds d) \<subseteq> {x. Xd x = d}"
simps_4: fixes var :: "nat"   and p :: "real mpoly"   and xs :: "real list" shows "convert_atom var (Neq p) xs = map_option NeqUni (convert_poly var p xs)"
invertible_modification_axioms: shows "invertible_modification (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G G \<Phi>\<^sub>G \<tau>\<tau>'.map\<^sub>0 \<tau>\<tau>'.map\<^sub>1 G.map\<^sub>0 (\<lambda>f. \<l>\<^sub>D\<^sup>-\<^sup>1[G f] \<cdot>\<^sub>D \<r>\<^sub>D[G f]) counit"
comm: fixes a :: "'a"   and b :: "'a" shows "a + b = b + a"
assms_2: shows "s\<^sub>2' = s\<^sub>1(r' \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r'')]), r'' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e))"
inner_quat_def: fixes x :: "quat"   and y :: "quat" shows "x \<bullet> y = quat.Re x * quat.Re y + Im1 x * Im1 y + Im2 x * Im2 y + Im3 x * Im3 y"
m: shows "max_Var_floatariths fas \<le> DIM('a) + length ramsch"
Rats_minus_iff: fixes a :: "'a" shows "(- a \<in> \<rat>) = (a \<in> \<rat>)"
disc_2: fixes x2 :: "'a ipt_iprange" shows "\<not> is_Src (Dst x2)"
monotone_case_prod_applyD: fixes orda :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> 'd"   and y :: "'e" assumes "monotone orda ordb (\<lambda>x. (case x of (x, xa) \<Rightarrow> f x xa) y)" shows "monotone orda ordb (\<lambda>(a, b). f a b y)"
singleton_insert_inj_eq': fixes a :: "'a"   and A :: "'a set"   and b :: "'a" shows "(insert a A = {b}) = (a = b \<and> A \<subseteq> {b})"
ide_compE: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a" assumes "category C"   and "partial_magma.ide C (C g f)" obtains "partial_magma.arr C (C g f)" and "partial_magma.arr C (C f g)" and "C g f = partial_magma.dom C f" and "C g f = partial_magma.cod C g"
bisim1_xcp_Some_not_caught: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'a \<Rightarrow> 'b"   and pc' :: "nat"   and d :: "nat" assumes "P,e,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "match_ex_table (compP f P) (cname_of h a) (pc' + pc) (compxE2 e pc' d) = None"
C: fixes \<mu> :: "'a" assumes "\<guillemotleft>\<mu> : f \<star> v \<Rightarrow> u\<guillemotright>" shows "trnl\<^sub>\<epsilon> u (trnl\<^sub>\<eta> v \<mu>) = \<mu>"
Tasked_Proof_Graph_axioms: shows "Tasked_Proof_Graph (curry (SOME f. bij f)) (\<lambda>_. id) (\<lambda>_. {}) Propositional_Formulas.closed subst (\<lambda>_. {}) (\<lambda>_. id) (Var undefined) antecedent consequent prop_rules [Fun imp [Fun and [A, B], C]] [Fun imp [A, Fun imp [B, C]]] task2_11.vertices task2_11.nodeOf task2_11.edges task2_11.vidx task2_11.inst"
word_set_nth_iff: fixes w :: "'b word"   and n :: "nat"   and b :: "bool" shows "(set_bit_class.set_bit w n b = w) = (w !! n = b \<or> size w \<le> n)"
compatible_iff_seq: fixes g :: "'b"   and f :: "'b" shows "(C'.domains g \<inter> C'.codomains f \<noteq> {}) = C'.seq g f"
psubset_eq: fixes A :: "'b set"   and B :: "'b set" shows "(A \<subset> B) = (A \<subseteq> B \<and> A \<noteq> B)"
eq_0_on_span: fixes b :: "'b set"   and x :: "'b" assumes "\<And>x. x \<in> b \<Longrightarrow> f x - g x = (0::'c)"   and "x \<in> vs1.span b" shows "f x - g x = (0::'c)"
normalize_quot_0_1: fixes x :: "'b" shows "normalize_quot (0::'b, x) = (0::'b, 1::'b)"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a' \<Rightarrow>\<^sub>C a\<guillemotright>"   and "C.iso (C.L f)" shows "C.iso f"
inf_top_right: fixes x :: "'a upred" shows "(x \<and> true) = x"
01: shows "{0..1} = {0..1 / 2} \<union> {1 / 2..1}"
shift_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {shift_type SLL, shift_type SRL, shift_type SRA}" shows "snd (shift_instr i s) = False"
nhdI: fixes U :: "'a set"   and m :: "'a set"   and x :: "'a" assumes "U \<subseteq> carrier"   and "m open"   and "x \<in> m"   and "m \<subseteq> U" shows "U \<in> nhds x"
y_aux: fixes y :: "nat"   and k :: "nat" shows "y \<le> Max_triangle_aux y k"
While: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "\<turnstile>\<^sub>2\<^sub>' {wp\<^sub>Q c Q} c {Q}"
col_cop__cop: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p" assumes "Coplanar A B C D"   and "C \<noteq> D"   and "Col C D E" shows "Coplanar A B C E"
inf1D2: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "(A \<sqinter> B) x" shows "B x"
poly_divide_filterlim_at_top: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "LIM x at_top. poly q x / poly p x :> if degree q < degree p then at 0 else if degree q = degree p then nhds (lead_coeff q / lead_coeff p) else if 0 < sgn_pos_inf q * sgn_pos_inf p then at_top else at_bot"
deg_H: fixes n :: "nat" assumes "0 < n" shows "degree (H $ n) < degree h"
subgroupI: fixes G :: "('a \<times> 'a) set set" assumes "G \<subseteq> carrier_local_ring_at"   and "zero_local_ring_at \<in> G"   and "\<And>g h. \<lbrakk>g \<in> G; h \<in> G\<rbrakk> \<Longrightarrow> add_local_ring_at g h \<in> G"   and "\<And>g. g \<in> G \<Longrightarrow> additive.invertible g"   and "\<And>g. g \<in> G \<Longrightarrow> - g \<in> G" shows "subgroup G carrier_local_ring_at add_local_ring_at zero_local_ring_at"
good_ruleset_normalize_rules: fixes rs :: "'b rule list"   and f :: "'b match_expr \<Rightarrow> 'b match_expr list" assumes "good_ruleset rs" shows "good_ruleset (normalize_rules f rs)"
lookup_except_singleton: fixes p :: "'c \<Rightarrow>\<^sub>0 'b"   and t :: "'c" shows "lookup (except p {t}) t = (0::'b)"
Suc_less_SucD: fixes m :: "nat"   and n :: "nat" assumes "Suc m < Suc n" shows "m < n"
r_cancel_one': fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier H"   and "a \<in> carrier H" shows "(x = a \<otimes>\<^bsub>H\<^esub> x) = (a = \<one>\<^bsub>H\<^esub>)"
assms_3: shows "ip \<in> kD (rt (\<sigma> sip)) \<and> sn \<le> nsqn (rt (\<sigma> sip)) ip \<and> (nsqn (rt (\<sigma> sip)) ip = sn \<longrightarrow> the (dhops (rt (\<sigma> sip)) ip) \<le> hops \<or> the (flag (rt (\<sigma> sip)) ip) = Aodv_Basic.inv)"
simps: fixes a1 :: "'v vertex"   and a2 :: "'v vertex" shows "edg a1 a2 = ((\<exists>x y. a1 = Vertex x \<and> a2 = Edge x y \<and> edge \<Delta> x y) \<or> (\<exists>x y. a1 = Edge x y \<and> a2 = Vertex y \<and> edge \<Delta> x y))"
oconf_upd_obj: fixes P :: "'b prog"   and h :: "heap"   and obj :: "obj"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields"   and fs' :: "Objects.fields" assumes "P,h \<turnstile> obj \<surd>"   and "h a = \<lfloor>(C, fs)\<rfloor>" shows "P,h(a \<mapsto> (C, fs')) \<turnstile> obj \<surd>"
simps_1: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and k :: "'c" shows "rbt_comp_lookup c rbt.Empty k = None"
Suc: shows "Suc m \<le> length (paid_A ! n)"
BuildSeq3_induct: fixes B :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and a :: "hf"   and a' :: "hf"   and a'' :: "hf"   and P :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool" assumes "BuildSeq3 B C s k a a' a''"   and "\<And>x x' x''. B x x' x'' \<Longrightarrow> P x x' x''"   and "\<And>x x' x'' y y' y'' z z' z''. \<lbrakk>C x x' x'' y y' y'' z z' z''; P y y' y''; P z z' z''\<rbrakk> \<Longrightarrow> P x x' x''" shows "P a a' a''"
univ_list_def: shows "univ_list = (SOME xs. distinct xs \<and> set xs = UNIV)"
splitAt_distinct_snd: fixes vs :: "'a list"   and ram1 :: "'a" assumes "distinct vs" shows "distinct (snd (splitAt ram1 vs))"
simps_2: fixes i :: "nat"   and j :: "nat" shows "[Suc i>..j] = (if j \<le> i then i # [i>..j] else [])"
OUTfromVCorrect1_data17: shows "OUTfromVCorrect1 data17"
rpsem_Assign_simps: fixes x :: "fin_string"   and \<theta> :: "trm"   and \<nu> :: "fin_string \<Rightarrow> real"   and b :: "fin_string \<Rightarrow> real" shows "([x := \<theta>]\<nu> \<downharpoonleft> b) = (\<exists>r. ([\<theta>]\<nu> \<down> r) \<and> b = \<nu>(x := r))"
fact_dvd: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "fact n dvd fact m"
transitive_closure_trans_3: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set"   and z :: "'b" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
a: shows "a \<in> (eintF (Tag \<sigma>) \<circ> singl) ` {b. eintT \<sigma> b}"
divide_lt1: fixes c :: "rat"   and a :: "'a"   and b :: "'a" assumes "c *R a < b"   and "c < 0" shows "(1 / c) *R b < a"
Y_def: fixes f :: "'c" shows "\<Phi>'.Y f \<equiv> Cop_S.Map (local.map f)"
span_eq_iff: fixes s :: "'d set" shows "(cspan s = s) = csubspace s"
set_upds_list: fixes S :: "stack_elem list" shows "set (upds_list S) = upds S"
AssertionStatImpRefl: fixes \<Psi> :: "'b" shows "\<Psi> \<hookrightarrow> \<Psi>"
main_method: shows "P \<turnstile> C sees M: []\<rightarrow>T = m in C"
F'_eq_F: fixes name :: "name" assumes "name \<in> Vs" shows "F' name = F name"
distrib: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "comm_semiring_ow U plus times"   and "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "times (plus a b) c = plus (times a c) (times b c)"
that: fixes \<chi> :: "'a \<Rightarrow> complex" assumes "character G \<chi>"   and "\<chi> x \<noteq> 1" shows "thesis"
starfun_o2: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "(\<lambda>x. (*f* f) ((*f* g) x)) = *f* (\<lambda>x. f (g x))"
S_X_generate: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<exists>f. pol_coeff S f \<and> x = polyn_expr R X (fst f) f"
ccBind_eq: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and v :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "CoCallAnalysis.ccBind ccExp v e\<cdot>(ae, G) = (if v--v\<notin>G \<or> \<not> isVal e then fup\<cdot>(ccExp e)\<cdot>(ae v) G|` fv e else fv e\<^sup>2)"
eval_dioph_div_mult: fixes ks :: "int list"   and n :: "int"   and xs :: "int list" shows "eval_dioph ks (map (\<lambda>x. x div n) xs) * n + eval_dioph ks (map (\<lambda>x. x mod n) xs) = eval_dioph ks xs"
lincomb_scalar_prod_right: fixes W :: "'a vec set"   and v :: "'a vec"   and a :: "'a vec \<Rightarrow> 'a" assumes "W \<subseteq> carrier_vec n"   and "v \<in> carrier_vec n" shows "v \<bullet> lincomb a W = (\<Sum>w\<in>W. a w * (v \<bullet> w))"
hX_def: fixes X :: "'a set"   and h :: "'c" shows "Bossy''.hX X h = {x \<in> X. Xh x = h}"
fold_congs_9: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme"   and v' :: "nat \<Rightarrow> Literal option"   and f :: "(nat \<Rightarrow> Literal option) \<Rightarrow> nat \<Rightarrow> Literal option"   and f' :: "(nat \<Rightarrow> Literal option) \<Rightarrow> nat \<Rightarrow> Literal option" assumes "r = r'"   and "getWatch2 r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "getWatch2_update f r = getWatch2_update f' r'"
prefs_wf'_1: fixes i :: "'agent" assumes "i \<in> agents" shows "total_preorder_on alts (R i)"
PER: shows "part_equiv (Restr (uconnected F) verts)"
transpose: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list" assumes "mat nr nc m" shows "mat nc nr (Matrix_Legacy.transpose nr m)"
cont_prodI: fixes orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<times> 'e \<Rightarrow> 'a"   and lubb :: "'e set \<Rightarrow> 'e"   and luba :: "'d set \<Rightarrow> 'd" assumes "monotone (rel_prod orda ordb) (\<le>) f"   and "\<And>x. cont lubb ordb Sup (\<le>) (\<lambda>y. f (x, y))"   and "\<And>y. cont luba orda Sup (\<le>) (\<lambda>x. f (x, y))"   and "class.preorder orda (mk_less orda)"   and "class.preorder ordb (mk_less ordb)" shows "cont (prod_lub luba lubb) (rel_prod orda ordb) Sup (\<le>) f"
zip_replicate2: fixes xs :: "'a list"   and n :: "nat"   and y :: "'b" shows "zip xs (replicate n y) = map (\<lambda>x. (x, y)) (take n xs)"
subgroup_is_group: assumes "Group.group (add_monoid M)" shows "Group.group (add_monoid M\<lparr>carrier := H\<rparr>)"
The_opt_def: fixes p :: "'a \<Rightarrow> bool" shows "The_opt p = (if \<exists>!x. p x then Some (The p) else None)"
update_correct: fixes m :: "'b"   and k :: "'a"   and v :: "'c" assumes "P m" shows "local.map_of (snd (run_state (update k v) m)) \<subseteq>\<^sub>m local.map_of m(k \<mapsto> v)"
compositor_is_natural_transformation: shows "transformation_by_components VV.comp (\<bullet>) HoSPN_SPN.map SPNoH.map (\<lambda>rs. CMP (fst rs) (snd rs))"
in_snapshot_def: fixes r :: "'h" shows "in_snapshot r = (\<lambda>s. black r s \<or> (\<exists>x. (x grey_protects_white r) s))"
map_cong0: fixes x :: "'b option"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>z. z \<in> set_option x \<Longrightarrow> f z = g z" shows "map_option f x = map_option g x"
not_inf_impl_terminatess: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate" assumes "\<not> AlternativeSmallStep.inf \<Gamma> cs css s" shows "\<Gamma>\<turnstile>cs,css \<Down> s"
hermitean_mult_real: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and k :: "real" assumes "hermitean H" shows "hermitean (cor k *\<^sub>s\<^sub>m H)"
assms_2: shows "\<guillemotleft>\<mu> : f \<star> src f \<Rightarrow> f\<guillemotright>"
simps_3: fixes \<A> :: "'a ltl set"   and \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" shows "\<A> \<Turnstile>\<^sub>P \<phi> and \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<and> \<A> \<Turnstile>\<^sub>P \<psi>)"
cInf_lower: fixes x :: "'b multiset"   and X :: "'b multiset set" assumes "x \<in> X"   and "subset_mset.bdd_below X" shows "Inf X \<subseteq># x"
PlusE: fixes u :: "'b + 'c"   and A :: "'b set"   and B :: "'c set" assumes "u \<in> A <+> B" obtains x :: "'b" where "x \<in> A" and "u = Inl x"   | y :: "'c" where "y \<in> B" and "u = Inr y"
assms_3: shows "A - B \<union> (B - A) \<in> null_sets M"
newman: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "wfP R\<inverse>\<inverse>"   and "\<And>a b c. \<lbrakk>R a b; R a c\<rbrakk> \<Longrightarrow> \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"   and "R\<^sup>*\<^sup>* a b"   and "R\<^sup>*\<^sup>* a c" shows "\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
IHF_3: fixes GS :: "'a literal list list"   and \<Gamma> :: "'a formula multiset" assumes "disj_of_clause FR, image_mset disj_of_clause (mset (FS @ GS)) + \<Gamma> \<Rightarrow>\<^sub>n" shows "image_mset disj_of_clause (mset (FS @ GS)) + \<Gamma> \<Rightarrow>\<^sub>n"
tendsto_at_botI_sequentially: fixes f :: "real \<Rightarrow> 'b"   and y :: "'b" assumes "\<And>X. filterlim X at_bot sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y" shows "(f \<longlongrightarrow> y) at_bot"
inject_1: fixes x11 :: "'a"   and x12 :: "'a process"   and y11 :: "'a"   and y12 :: "'a process" shows "(Action x11 x12 = Action y11 y12) = (x11 = y11 \<and> x12 = y12)"
complete_lattice_ccpo_dual: shows "class.ccpo Inf (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x)"
lemm: fixes A :: "nat set"   and x :: "nat"   and y :: "nat" assumes "card {i \<in> A. i < x} = card {i \<in> A. i < y}"   and "x \<in> A"   and "y \<in> A" shows "x = y"
fps_nth_of_nat: fixes c :: "nat"   and n :: "nat" shows "of_nat c $ n = (if n = 0 then of_nat c else (0::'a))"
contI2: fixes f :: "'d \<Rightarrow> 'e" assumes "monofun f"   and "\<And>Y. \<lbrakk>chain Y; chain (\<lambda>i. f (Y i))\<rbrakk> \<Longrightarrow> f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. f (Y i))" shows "cont f"
beta_C_meta_2: fixes \<phi> :: "\<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<o>"   and v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" assumes "IsProperInXY \<phi>" shows "[\<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<phi> (x\<^sup>P) (y\<^sup>P)),x\<^sup>P,y\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P) (y\<^sup>P) in v]"
indices_Deliver_Delete: fixes i :: "'c"   and n :: "'c" shows "indices [Deliver (i, Delete n)] = []"
red_setI: fixes f :: "'a \<Rightarrow>\<^sub>0 'c"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "f \<in> F"   and "punit.red_single p q f t" shows "punit.red F p q"
set_def: fixes s' :: "'a" shows "State_Monad.set s' = State (\<lambda>_. ((), s'))"
div_rat_poly_step_def: fixes q :: "'a poly" shows "div_rat_poly_step q = (\<lambda>a (s, r). let b = coeff (pCons a r) (degree q) / lead_coeff q in (pCons b s, pCons a r - smult b q))"
chop_App_Hd: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" assumes "is_Hd s" shows "chop (App s t) = t"
inline1_try_gpv: fixes gpv :: "('c, 'call, 'ret) gpv"   and f :: "'c \<times> 's \<Rightarrow> 'd"   and x :: "'c"   and z :: "'d"   and s :: "'s" assumes "\<I> \<turnstile>g gpv \<surd>"   and "pfinite_gpv \<I> gpv"   and "\<And>s. I s \<Longrightarrow> f (x, s) = z"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> colossless_gpv \<I>' (callee s x)"   and "I s" shows "map_spmf (map_sum f id) (inline1 callee (TRY gpv ELSE Done x) s) = TRY map_spmf (map_sum f (\<lambda>(out, c, rpv). (out, c, \<lambda>input. TRY rpv input ELSE Done x))) (inline1 callee gpv s) ELSE return_spmf (Inl z)"
f_def: fixes fs :: "'b list"   and i :: "nat" shows "??.Newton_Interpolation.f fs i = fs ! i"
abelian_group_hom_axioms_def: fixes G :: "('c, 'd) ring_scheme"   and H :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "abelian_group_hom_axioms G H h \<equiv> group_hom (add_monoid G) (add_monoid H) h"
y_1: shows "y = a + v *\<^sub>R (b - a)"
llast_conv_lnth: fixes xs :: "'c llist"   and n :: "nat" assumes "llength xs = eSuc (enat n)" shows "llast xs = xs $ n"
plus_period: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "periodic_fun_simple' f" shows "f (x + (1::'b)) = f x"
Nml_implies_Arr: fixes t :: "'a E.term" assumes "E.Nml t" shows "E.Arr t"
simps: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat \<Rightarrow> nat option"   and a3 :: "exp list"   and a4 :: "nat list" shows "a1;a2 \<turnstile>+ a3 : a4 = ((\<exists>CT \<Gamma>. a1 = CT \<and> a2 = \<Gamma> \<and> a3 = [] \<and> a4 = []) \<or> (\<exists>CT \<Gamma> e0 C0 es Cs. a1 = CT \<and> a2 = \<Gamma> \<and> a3 = e0 # es \<and> a4 = C0 # Cs \<and> CT;\<Gamma> \<turnstile> e0 : C0 \<and> CT;\<Gamma> \<turnstile>+ es : Cs))"
tauChainAddTau_1: fixes P :: "pi"   and P' :: "pi"   and P'' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "P' \<longmapsto> \<tau> \<prec> P''" shows "P \<Longrightarrow>\<^sub>\<tau> P''"
order_iff: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b \<subseteq># a) = (a = a \<union># b)"
inv_def: fixes c :: "'a \<Rightarrow> 'b" shows "Abstract_Linear_Poly.inv c \<equiv> finite {v. c v \<noteq> (0::'b)}"
snd_diag_snd: shows "snd \<circ> ((\<lambda>x. (x, x)) \<circ> snd) = snd"
ss_NS_not_UNIV: assumes "ssimple" shows "NS \<noteq> UNIV"
to_int_poly_i: fixes f :: "'i list"   and g :: "'a mod_ring poly" assumes "poly_rel f g" shows "to_int_poly_i ff_ops f = to_int_poly g"
ncols_mult: fixes A :: "'b matrix"   and B :: "'b matrix" shows "ncols (A * B) \<le> ncols B"
shadow_root_ptr_casts_commute2: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "cast (cast shadow_root_ptr) = Some shadow_root_ptr"
assms: fixes Pd :: "'d \<Rightarrow> ('x \<times> 'x) set"   and Ch :: "'h \<Rightarrow> 'x set \<Rightarrow> 'x set"   and Pd' :: "'d \<Rightarrow> ('x \<times> 'x) set"   and ds' :: "'d set" assumes "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch"   and "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (override_on Pd Pd' ds') Ch"   and "ds' \<subseteq> ds"   and "ds' \<noteq> {}"   and "\<forall>d\<in>ds'. \<exists>y\<in>\<phi> (override_on Pd Pd' ds') Ch ds. y \<in> AboveS (Pd d) (dX (\<phi> Pd Ch ds) d)" shows "False"
countable_or_card_of: fixes A :: "'b set" assumes "countable A" shows "finite A \<and> |A| <o |UNIV| \<or> infinite A \<and> |A| =o |UNIV|"
ab_specD2: fixes ab :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" assumes "ab_spec ab" shows "ab gs bs [] data = bs"
counit_simps_1: fixes a :: "'c" assumes "C.obj a" shows "C.iso (\<epsilon>'.counit a)"
set_spmf_of_set_finite: fixes A :: "'c set" assumes "finite A" shows "set_spmf (spmf_of_set A) = A"
apply_cltn2_abs_abs_in_S: fixes v :: "(real, 3) vec"   and J :: "((real, 3) vec, 3) vec" assumes "v \<noteq> 0"   and "invertible J" shows "(apply_cltn2 (proj2_abs v) (cltn2_abs J) \<in> S) = (v \<bullet> (J ** M ** Finite_Cartesian_Product.transpose J *v v) = 0)"
open_UN: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'a set" assumes "\<forall>x\<in>A. open (B x)" shows "open (\<Union> (B ` A))"
extend_for_any_seq: fixes R1 :: "(sequent list \<times> sequent) set"   and R2 :: "(sequent list \<times> sequent) set"   and R3 :: "(sequent list \<times> sequent) set"   and R :: "(sequent list \<times> sequent) set"   and r :: "sequent list \<times> sequent"   and S :: "sequent" assumes "R1 \<subseteq> upRules \<and> R2 \<subseteq> nprovRules \<and> R3 \<subseteq> provRules"   and "R = Ax \<union> R1 \<union> R2 \<union> R3"   and "r \<in> R" shows "extendRule S r \<in>  R*"
n_2: fixes i :: "nat" assumes "i < n" shows "\<not> (enat i < llength P \<and> P $ i \<notin> A - W)"
cp0: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3)"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "(case a1 of (a, b, c) \<Rightarrow> f self a b c) \<tau> = (case (case a1 of (a1, a2, a3) \<Rightarrow> \<lambda>\<tau>. (\<lambda>_. a1 \<tau>, \<lambda>_. a2 \<tau>, \<lambda>_. a3 \<tau>)) \<tau> of (a, b, c) \<Rightarrow> f (\<lambda>_. self \<tau>) a b c) \<tau>"